CHAPTER 3 PULSE MODULATION

CHAPTER 3 PULSE MODULATION
Digital Communication Systems 2012 R. Sokullu 1/68

Digital Communication Systems 2012
Outline 3.1 Introduction 3.2 Sampling Process 3.3 Pulse Amplitude Modulation 3.4 Other Forms of Pulse Modulation 3.5 Bandwidth-Noise Trade-off 3.6 The Quantization Process Digital Communication Systems 2012 R.Sokullu 2/68

3.1 Introduction This chapter is a transitional chapter between analog and digital modulation techniques. In CW modulation, as we have studied in chapter 2, one parameter of the sinusoidal carrier wave is continuously varied in accordance with a given message signal. In the case of pulse modulation we have a pulse train and some parameter of the pulse train is varied in accordance with the message signal. Digital Communication Systems 2012 R.Sokullu 3/68

CommSystems 1 – Analog Communication Techniques
In the first part of Communication Systems we studied transmission techniques of analog waveforms (analog sources) over analog signals (lines). Why is modulation necessary? What types of modulation did we study? When we studied a specific modulation type what were the specific subjects we discussed? Digital Communication Systems 2012 R.Sokullu 4/68

CommSystems 2 – Digital Communication Techniques
In the second part we have two major topics analog waveforms (analog sources) transmission using baseband signals digital waveforms (digital sources) transmission using band-pass signals Digital Communication Systems 2012 R.Sokullu 5/68

Why digital? Digital approximation of analog signals can be made as precise as required Low cost of digital circuits Flexibility of digital approach – possibility of combining analog and digital sources for transmission over digital lines Increased efficiency – source coding/channel coding separation Digital Communication Systems 2012 R.Sokullu 6/68

Goals of this course: To study digital communication systems and their conceptual basis in information theory To study how analog waveforms can be converted to digital signals (PCM) Compute spectrum of digital signals Examine effects of filtering – how does filtering affect the ability to recover digital information at the receiver. filtering produces ISI in the recovered signal Study how to multiplex data from several digital bit streams into one high speed digital stream for transmission over a digital system (TDM) Digital Communication Systems 2012 R.Sokullu 7/68

Motivation and Development Digital transmission –1960s Real application – after 1970s developments in solid state electronics, micro-electronics, large scale integration all common information sources are inherently analog Historical steps Sampled analog sources transmitted using analog pulse modulation (PAM, PPM) Samples are quantized to discrete levels (PCM, DM) Conversion from analog and transmission were implemented as a single step Today Layered approach – different steps are distinguished and separately optimized (source coding and channel coding) Digital Communication Systems 2012 R.Sokullu 8/68

We distinguish between: analog pulse modulation a periodic pulse train is used as a carrier wave; a parameter of that train (amplitude, duration, position) is varied in a continuous manner in accordance with the corresponding sample value of the message signal; information is transmitted basically in analog form, but at discrete times. digital pulse modulation message represented in a discrete way in both time and amplitude; sequence of coded pulses is transmitted. Digital Communication Systems 2012 R.Sokullu 9/68

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Source Coding Problem of coding: efficient representation of source signals (speech waveforms, image waveforms, text files) as a sequence of bits for transmission over a digital network Paired problem of source decoding – conversion of received bit sequence (possibly corrupted) into a more-or-less faithful replica of the original Digital Communication Systems 2012 R.Sokullu 11/68

Channel Coding Problem of the efficient transmission of a sequence of bits through a lower layer channel 4 KHz telephone channel, wireless channel Recovery at the channel output in the remote receiver despite distortions Digital Communication Systems 2012 R.Sokullu 12/68

Why separate source and channel coding? Basic theorem of information theory: If a source signal can be communicated through a given point-to-point channel within some level of distortion (by any means) then the separate source and channel coding can also be designed to stay within the same limits of distortion. WHY then…(delay, complexity…) Pros and cons? Does it always hold true? Digital Communication Systems 2012 R.Sokullu 13/68

Shannon and the Channel Coding Theorem Channel coding can help reduce the error probability without reducing the data rate Date rate depends on the channel itself – channel capacity Channel bandwidth W, input power P, noise power then the channel capacity in bits is: Digital Communication Systems 2012 R.Sokullu 14/68

Digital Interface Interface between source coding/channel coding – issues continuity, rate etc. continuous sources packet sources complex combinations Here: min number of bits from source and max transmission speed over channel source coder rate = channel encoder rate (source-channel coding theorem) protocols discussed in details in Data Communications course Digital Communication Systems 2012 R.Sokullu 15/68

3.2 Sampling Process Sampling converts an analog signal into a corresponding sequence of samples that are uniformly distributed in time. Proper selection of the sampling rate is very important because it determines how uniquely the samples would represent the original signal. It is determined according to the so called sampling theorem. Digital Communication Systems 2012 R.Sokullu 17/68

The model: we consider an arbitrary signal g(t) with finite energy, specified for all time we sample the signal instantaneously and at an uniform rate, once every Ts seconds we obtain an infinite sequence of samples spaced Ts seconds apart; they are denoted by [g(nTs)], where n can take all possible integer values Ts is referred to as the sampling period, and fs=1/Ts is the sampling rate. let gδ(t) denote the signal obtained by individually weighting the elements of a periodic sequence of delta functions spaced Ts seconds apart by the numbers [g(nTs)]. Digital Communication Systems 2012 R.Sokullu 18/68

The sampling process. (a) Analog signal. (b) Instantaneously sampled version of the analog signal. Figure 3.1 Digital Communication Systems 2012 R.Sokullu 19/68

For the signal gδ(t), called the ideal sampled signal, we have the following expression : As the idealized delta function has unit area, the multiplication factor g(nTs) can be considered as “mass” assigned to it (samples are “weighted”); A delta function weighted in this manner is approximated by a rectangular pulse of duration Δt and amplitude g(nTs)/Δt. Digital Communication Systems 2012 R.Sokullu 20/68

Knowing that the uniform sampling of a continuous- time signal of finite energy results into a periodic spectrum with a period equal to the sampling rate using the FT gδ(t) can be expressed as: So if we take FT on both sides of (3.1) we get: discrete time Fourier transform Digital Communication Systems 2012 R.Sokullu 21/68

The relations derived up to here apply to any continuous time signal g(t) of finite energy and infinite duration. If the signal g(t) is strictly band-limited, with no components above W Hz, then the FT G(f) of g(t) will be zero for |f| ≥ W. Digital Communication Systems 2012 R.Sokullu 22/68

(a) Spectrum of a strictly band-limited signal g(t). (b) Spectrum of the sampled version of g(t) for a sampling period Ts = 1/2 W. Figure 3.2 Digital Communication Systems 2012 R.Sokullu 23/68

For a sampling period Ts=1/2 W after substitution in 3.3 we get the following expression: and using 3.2 for the FT of gδ(t) we can also write: m=0 Digital Communication Systems 2012 R.Sokullu 24/68

and for the conditions specified about f we get: and when we substitute (3.4) and (3.6) we get: Digital Communication Systems 2012 R.Sokullu 25/68

Conclusion: 1. If the sample values g(n/2W) of an analog signal g(t) are specified for all n, then the FT G(f) of the signal is uniquely determined by using the discrete-time FT of equation (3.7). 2. Because g(t) is related to G(f) by the inverse FT, the signal g(t) is itself uniquely determined by the sample values g(n/2W) for -∞ < n <+∞. Digital Communication Systems 2012 R.Sokullu 26/68

Second part: reconstructing the signal from the samples We substitute equation (3.7) in the inverse FT formula and after some reorganizing we get: which after integration ends to be: Digital Communication Systems 2012 R.Sokullu 27/68

This is an interpolation formula for reconstructing the original signal g(t) from a sequence of sample values [g(n/2W)]. The sinc function sinc(2Wt) is playing the role of interpolation function. Each sample is multiplied by a suitably delayed version of the interpolation function and all the resulting waveforms are summed up to obtain g(t). Digital Communication Systems 2012 R.Sokullu 28/68

Sampling Theorem 1. A band-limited signal of finite energy, which has no frequency components higher than W Hz, is completely described by specifying the values of the signal at instants of time separated by 1/2W (means that sampling has to be done at a rate twice the highest frequency of the original signal). 2. A band-limited signal of finite energy, which has no frequency components higher than W Hz, may be completely recovered from a knowledge of its samples taken at the rate of 2W samples per second. Digital Communication Systems 2012 R.Sokullu 29/68

Note: The sampling rate of 2W for a signal of bandwidth W Hz, is called the Nyquist rate; Its reciprocal 1/2W (seconds) is called the Nyquist interval; Digital Communication Systems 2012 R.Sokullu 30/68

The derivations of the sampling theorem so far were based on the assumption that the signal g(t) is strictly band limited. Practically – not strictly band-limited; the result is under sampling so some aliasing is produced by the sampling process. Aliasing is the phenomenon of a high- frequency component in the spectrum of the signal taking on the identity of a lower frequency in the spectrum of its sampled version. Digital Communication Systems 2012 R.Sokullu 31/68

(a) Spectrum of a signal. (b) Spectrum of an undersampled version of the signal exhibiting the aliasing phenomenon. Figure 3.3 Digital Communication Systems 2012 R.Sokullu 32/68

Practically there are two possible engineering solutions: prior to sampling, a low-pass anti-aliasing filter is used to attenuate the high-frequency components that are not essential to the information baring signal. the filtered signal is sampled at a rate slightly higher than the Nyquist rate. Note: This also makes the design of the reconstructing filter easier. Digital Communication Systems 2012 R.Sokullu 33/68

Anti-alias filtered spectrum of an information-bearing signal. (b) Spectrum of instantaneously sampled version of the signal, assuming the use of a sampling rate greater than the Nyquist rate. (c) Magnitude response of reconstruction filter. Figure 3.4 Digital Communication Systems 2012 R.Sokullu 34/68

The reconstruction filter is low-pass, pass-band –W to +W. The transition band of the filter is fs- W where fs is the sampling rate. Digital Communication Systems 2012 R.Sokullu 35/68

3.3 Pulse Amplitude Modulation Definition: In Pulse Amplitude Modulation (PAM) the amplitudes of regularly spaced pulses are varied in accordance with the corresponding sample values of the continuous message signal; Note: Pulses can be rectangular or some other form. Digital Communication Systems 2012 R.Sokullu 37/68

Flat-top samples, representing an analog signal. Figure 3.5 Digital Communication Systems 2012 R.Sokullu 38/68

PAM Steps in realizing PAM: Instantaneous sampling of the message signal every Ts seconds, with sampling rate fs chosen according to the sampling theorem. Lengthening the duration of each sample to obtain a constant value of T (duration of pulses). These two are known as “sample and hold”. Question is: how long should be the pulses (T)? Digital Communication Systems 2012 R.Sokullu 39/68

Assume: s(t) sequence of flat-top pulses generated as described. where Ts is the sampling period, m(nTs) is the sample value at time t=nTs standard rectangular pulse is represented by: by definition the instantaneously sampled version of m(t) is: time-shifted delta function Digital Communication Systems 2012 R.Sokullu 40/68

after convolution and applying the sifting property of the delta function we get: Digital Communication Systems 2012 R.Sokullu 41/68

The result in the previous slide means that (compare and 3.14) the PAM signal s(t) is mathematically represented by 3.15: Digital Communication Systems 2012 R.Sokullu 42/68

After taking FT on both sides we get: Using formula 3.2 for the relation between Mδ(f) and M(f), the FT of the original message m(t) we can write: Finally, after substitution of 3.16 into 3.17 we get which represents the FT of the PAM signal s(t). Digital Communication Systems 2012 R.Sokullu 43/68

Second part: recovery procedure assume that the message is limited to bandwidth W and the sampling rate is fs which is higher than the Nyquist rate. pass s(t) through a low-pass filter whose frequency response is defined in 3.4c the result, according to 3.18 is M(f)H(f), which is equal to passing the original signal m(t) through another low-pass filter with frequency response H(f). Fig. 3.4 Digital Communication Systems 2012 R.Sokullu 44/68

To determine H(f) we use the FT of a rectangular pulse, plotted on fig. 3.6a and 3.6b: By using flat-top samples to generate a PAM signal we introduce amplitude distortion and delay of T/2 This distortion is known as the aperture effect. This distortion is corrected by the use of an equalizer after the low-pass filters to compensate for the aperture effect. The magnitude response of the equalizer is ideally: Digital Communication Systems 2012 R.Sokullu 45/68

Rectangular pulse h(t). (b) Spectrum H(f), made up of the magnitude |H(f)|, and phase arg[H(f)] Figure 3.6 Digital Communication Systems 2012 R.Sokullu 46/68

Conclusion on PAM: 1. Transmission of a PAM signal imposes strict requirements on the magnitude and phase responses of the channel, because of the relatively short duration of the transmitted pulses. 2. Noise performance can never be better than a baseband signal transmission. 3. PAM is used only for time division multiplexing. Later on for long distance transmission another subsequent pulse modulation is used. Digital Communication Systems 2012 R.Sokullu 47/68

3.4 Other Forms of Pulse Modulation Rough comparison between CW modulation and pulse modulation shows that latter inherently needs more bandwidth. This bandwidth can be used for improving noise performance. Such additional improvement is achieved by representing the sample values of the message signal by some other parameter of the pulse (different than amplitude): pulse duration (width) modulation (PDM) – samples are used to vary the duration of the individual pulses. pulse-position modulation (PPM) – position of the pulse, relative to its un-modulated time of occurrence in accordance with the message signal. Digital Communication Systems 2012 R.Sokullu 49/68

Illustrating two different forms of pulse-time modulation for the case of a sinusoidal modulating wave. (a) Modulating wave. (b) Pulse carrier. (c) PDM wave. (d) PPM wave. Figure 3.8 Digital Communication Systems 2012 R.Sokullu 50/68

Comparison: 1. In PDM long pulses require more power, so PPM is more effective. 2. Additive noise has no effect on the position of the pulse if it is perfectly rectangular (ideal) but in reality pulses are not so PPM is affected by channel noise. 3. As in CW systems the noise performance and comparison can be done using the output signal-to-noise ratio or the figure of merit. 4. Assuming the average power of the channel noise is small compared to the peak pulse power, the figure of merit for a PPM system is proportional to the square of the transmission bandwidth BT, normalized with respect to the message bandwidth W. 5. In bad noise conditions the PPM systems suffer a threshold of its own – loss of wanted message signal. Digital Communication Systems 2012 R.Sokullu 51/68

3.5 Bandwidth-Noise Trade-Off As far as the analog pulse modulation schemes are concerned the pulse position modulation exhibits optimum noise performance. In comparison with CW modulation schemes it is close to the FM systems. both systems have a figure of merit proportional to the square of the transmission bandwidth BT normalized with respect to the message bandwidth. both systems exhibit a threshold effect as the signal-to- noise ratio is reduced. can we do better – yes but not with analog methods…. Digital Communication Systems 2012 R.Sokullu 53/68

what is required is discrete representation in both time and amplitude. discrete in time – sampling discrete in amplitude – quantization Digital Communication Systems 2012 R.Sokullu 54/68

3.6 Quantization Process For a continuous signal (voice, music) the samples have a continuous amplitude range. But humans can detect only finite intensity differences So an original signal can be approximated, without loss of perception, by a signal constructed of discrete amplitudes selected on a min error basis. This is the basic condition for the existence of pulse code modulation. Digital Communication Systems 2012 R.Sokullu 56/68

Definition: Amplitude quantization is defined as the process of transforming the sample amplitude m(nTs) of a message signal m(t) at time t=nTs into a discrete amplitude of v(nTs) taken from a finite set of possible amplitudes. We assume that the quantization process is memoryless and instantaneous. (This means that the transformation at time t is not affected by earlier or later sample values.) Digital Communication Systems 2012 R.Sokullu 57/68

Description of a memoryless quantizer. Figure 3.9 Digital Communication Systems 2012 R.Sokullu 58/68

Types of quantizers based on the way representation values are distributed and positioned around the origin: unifrom – equally spaced representation levels non-uniform – non-equally; considered later mid-read – origin lies in the middle of a read; mid-rise – origin lies in the middle of the rising part of the staircase graph symmetric about the origin Digital Communication Systems 2012 R.Sokullu 59/68

Two types of quantization: (a) midtread and (b) midrise. Figure 3.10 Digital Communication Systems 2012 R.Sokullu 60/68

Quantization Noise Definition: The error caused by the difference between the input signal m and the output signal v is referred to as quantization noise. Digital Communication Systems 2012 R.Sokullu 61/68

Illustration of the quantization process. Figure 3.11 Digital Communication Systems 2012 R.Sokullu 62/68

The model Assume: input value m, which is the sample value of a zero-mean RV M; output value v which is the sample value of a RV V; quantizer g(*) that maps the continuous RV M into a discrete RV V; respective samples of m and v are connected with the following relation: or Digital Communication Systems 2012 R.Sokullu 63/68

We are trying to evaluate the quantization error Q. zero mean because the input is zero mean for the output signal-to-noise (quantization) ratio we need the mean square value of the quantization error Q. the amplitude of m varies (-mmax, mmax); then for uniform quantizer the step size is given by: with L being the total number of representation levels; for uniform quantizer the error is bounded by –Δ/2≤q≤Δ/2 if step size is small Q is uniformly distributed (L large) Digital Communication Systems 2012 R.Sokullu 64/68

as mean is zero, variance is: Digital Communication Systems 2012 R.Sokullu 65/68

hidden in Δ is the number of levels used, which directly influences the error. typically an L-ary number k, denoting the kth representation level of the quantizer is transmitted to the receiver in binary form. Let R denote the number of bits per sample used in the binary code. Digital Communication Systems 2012 R.Sokullu 66/68

so for the step size we get and for the variance If P denotes the average power of the message signal m(t) we can find the output signal-to-noise ratio as: Digital Communication Systems 2012 R.Sokullu 67/68

Conclusion: The output SNR of the quantizer increases exponentially with increasing the number of bits per sample, R. Increasing R means increase in BT. So, using binary code for the representation of a message signal provides a more efficient method for the trade-off of increased bandwidth for improved noise performance than either FM or PPM. Note: FM and PPM are limited by receiver noise, while quantization is limited by quantization noise. Digital Communication Systems 2012 R.Sokullu 68/68

CHAPTER 3 PULSE MODULATION
Digital Communication Systems 2012 R.Sokullu 69/61

Digital Communication Systems 2012 R.Sokullu
Outline 3.7 Pulse Code Modulation 3.8 Noise in PCM Systems 3.9 Time Division Multiplexing 3.10 Digital Multiplexers 3.11 Modifications of PCM Digital Communication Systems 2012 R.Sokullu 70/61

3.7 Pulse Code Modulation This part deals with the most basic form of digital modulation. It is based on the two main processes we have studied - the sampling process and the quantization process. Definition: Pulse Code Modulation is a technique where the message signal is represented by a sequence of coded pulses. It realizes digital representation of the signal both time-wise and amplitude-wise. Digital Communication Systems 2012 R.Sokullu 71/61

PCM essentially an analog-to-digital conversion (delta modulation (DM) and differential pulse code modulation (DPCM)); special – information contained in the instantaneous sample is represented by digital words in a serial bit stream. Transmitter sampling quantization (A/DC) encoding (A/DC) Receiver regeneration decoding reconstruction Digital Communication Systems 2012 R.Sokullu 72/61

The basic elements of a PCM system. Figure 3.13 Digital Communication Systems 2012 R.Sokullu 73/61

PCM Transmission System Digital Communication Systems 2012 R.Sokullu 74/61

Sampling train of narrow rectangular pulses > 2W (sampling theorem) low-pass filter – anti-aliasing effect result = limited number of discrete values per second Digital Communication Systems 2012 R.Sokullu 75/61

Quantization uniform law (described in sec.3.6) non-uniform – (voice applications); step size increases in accordance with input-output amplitude separation from origin compressor + uniform quantizer µ-law (m and v – normalized I/O voltages) µ-law - |m| >>1 – logarithmic; |m| << 1 – linear Digital Communication Systems 2012 R.Sokullu 76/61

Compression laws. (a) -law. (b) A-law. Figure 3.14 Digital Communication Systems 2012 R.Sokullu 77/61

A-law Digital Communication Systems 2012 R.Sokullu 78/61

Transmission side - Encoding Aim – robust to noise, interference and channel impairments (see Table 3.2/204) line codes differential codes discrete set of values – appropriate signal binary codes – 1 and 0 (resistant to high noise ratio) – 256 q. levels – 8 bit code word ternary codes - 1, 0 and -1 Digital Communication Systems 2012 R.Sokullu 79/61

Line codes for the electrical representations of binary data. (a) Unipolar non-return-to-zero (NRZ) signaling (on-off signalling). (b) Polar NRZ signaling. (c) Unipolar return-to-zero (RZ) signaling. (d) Bipolar RZ signaling. (e) Split-phase or Manchester code. Figure 3.15 Digital Communication Systems 2012 R.Sokullu 80/61

Bandwidth of PCM Signals What is the spectrum of a PCM data waveform For PAM – obtained as a function of the spectrum of the input analog signal, because PAM is a linear function of the signal PCM is non-linear function of the input analog signal Spectrum is not directly related to the spectrum of the input analog signal Bandwidth depends on: bit rate and pulse shape used to represent the data where n is the number of bits in the PCM word, sampling frequency. For no aliasing, (B is the analog signal bandwidth). Dimensionality theorem gives the bounds: Digital Communication Systems 2012 R.Sokullu 81/61

Bandwidth of PCM Signals Min bandwidth is for the case of Exact bandwidth depends on the type of line encoding used (unipolar NRZ, polar NRZ, bipolar RZ etc. Next slides provide information of bandwidth and power requirements for different line encoding schemes. For rectangular pulses first null bandwidth is: so lower bound for PCM is Digital Communication Systems 2012 R.Sokullu 82/61

Bandwidth of PCM Signals Finally, bandwidth for PCM signals in the case where sampling is higher than , is significantly higher than the corresponding analog signal it represents. Digital Communication Systems 2012 R.Sokullu 83/61

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Power spectra of line codes: Assumptions: 2. Average power is normalized to unity 1. Symbols 1 and 0 are equiprobable 3. Frequency is normalized to the bit rate 1/Tb Figure 3.16a (a) Unipolar NRZ signal. Disadvantages – DC component; power spectra – not 0 at 0 freq. Digital Communication Systems 2012 R.Sokullu 85/61

Average power is normalized to unity Figure 3.16b Frequency is normalized to the bit rate 1/Tb (b) Polar NRZ signal. Disadvantages – large power near zero frequency Digital Communication Systems 2012 R.Sokullu 86/61

Figure 3.16c (c) Unipolar RZ signal. Advantages – presence of delta function at f=0, 1/Tb- used for sync Disadvantage – 3dB more power polar RZ for same error probability Digital Communication Systems 2012 R.Sokullu 87/61

Figure 3.16d (d) Bipolar RZ signal. Advantages – no DC component; bipolar AMI Digital Communication Systems 2012 R.Sokullu 88/61

Figure 3.16e (e) Manchester-encoded signal. Advantages – no DC; insignificant low-frequency components Digital Communication Systems 2012 R.Sokullu 89/61

Differential Codes encoding based on signal transitions reference signal (1) is necessary Figure 3.17 Digital Communication Systems 2012 R.Sokullu 90/61

Transmission Path - Regeneration PCM advantage – control effects of noise and distortion PCM signal – reconstructed by a series of regenerative repeaters along the transmission route functions: equalization – reshaping, compensates for noise and distortion timing – circuitry to provide a periodic pulse train for determining sampling instants decision making – comparison to a predetermined threshold Note: Occasional wrong decisions = bit errors Digital Communication Systems 2012 R.Sokullu 91/61

Regeneration Possible problems: Noise and interference on the channel can add resulting in wrong decisions = bit errors Spacing between pulses can deviate from originally assigned = jitter Digital Communication Systems 2012 R.Sokullu 92/61

Block diagram of regenerative repeater. Figure 3.18 Digital Communication Systems 2012 R.Sokullu 93/61

Receiving side - Decoding Receiver side functions regeneration regrouping into code-words decoding Decoding: generating a pulse the amplitude of which is the linear sum of all pulses in the code word, with each pulse being weighted by its place value (20, 21,…2R-1) Digital Communication Systems 2012 R.Sokullu 94/61

Filtering Final operation – after decoder low-pass reconstruction filter with bandwidth W (message bandwidth). If transmission path is error free the recovered signal has: no noise from channel only distortion - quantization Digital Communication Systems 2012 R.Sokullu 95/61

3.8. Noise Considerations in PCM Systems Two major sources: channel noise quantization noise – signal dependent Digital Communication Systems 2012 R.Sokullu 97/61

Channel and Quantization Noise Channel Noise Introduces bit errors Fidelity – average probability of symbol errors (probability that the reconstructed symbol differ from the transmitted binary symbol); in BER (equal or weighted). Modeling - AWGN; reduce distance between repeaters; performance dependent on quantization noise Quantization noise –presented before; design stage Digital Communication Systems 2012 R.Sokullu 98/61

Error Threshold BER due to AWGN depends on Eb/N0 – ratio of the transmitted signal energy per bit Eb, to the noise spectral density N0. Table 3.3 – different behavior below and above 11 dB. (compare to dB for high quality speech transmission with AM). No error accumulation – regeneration Digital Communication Systems 2012 R.Sokullu 99/61

3.9. Time Division Multiplexing Figure 3.19 Digital Communication Systems 2012 R.Sokullu 101/61

Concept 1. Restricting each input by low-pass anti-aliasing filter 2. Commutator – takes sample from each input message (f > 2W); interleave samples in a frame Ts; 3. Pulse modulator – transformation for transmission over common channel 4. Pulse demodulator 5. Decommutator – synchronized with the commutator Digital Communication Systems 2012 R.Sokullu 102/61

Synchronization TDM - Easy to add and drop sources Pulses duration considerations time interval limited by the sampling rate (reciprocal) more users – shorter pulses – difficult to generate; highly influenced by impairments upper limit of number of independent sources Transmitter-receiver clock sync – very important – two local clocks separate code element or pulse at the end of a frame orderly procedure for detecting sync pulses – searching procedure Digital Communication Systems 2012 R.Sokullu 103/61

Example: The T1 System 24 voice channels; separate pairs of wires; regeneration every 2 km; basic to the North American Digital Switching Hierarchy Voice signal (300 – 3100 Hz) – low pass filter (cutoff frequency 3.1 kHz) – Nyquist sampling rate = 6.2 kHz – actual sampling rate 8 kHz Companding - µ-law; µ = 255; 15 piece linear segment for approximating the logarithmic characteristic; 1a, 2a, 3a … segments above x, 1b, 2b, 3b,…below x; 14 segments, each segment contains 16 uniform decision levels for segment 0 – quantizer inputs are: ±1,±3, …±31 and the outputs are 0, ±1, ….±15; for segment 1a and 1b the decision level quantizer inputs are: ±31, ±35, …±95 and the outputs are ±16, ±17,…±31 and so on for the other linear segments (up to 7a and 7b). Finally we have equally spacing on the y axis corresponding to non-equally spaced inputs on the x axis (different step for different segment); Total representation levels: X16 = 255 for the 15 segment companding characteristic; Digital Communication Systems 2012 R.Sokullu 104/61

Each of the 24 voice channels uses binary code with 8-bit word. first bit – 1 (positive voice input), 0 (negative voice input) bits 2 – 4 – identify particular segment last 4 bits – actual representation level (16 levels) Frames for 8 kHz, each frame occupies a period of 125 µs contains 24 X 8 =192 bit words; 1 bit for sync = 193 bits bit duration = µs (125µs/193bits); transmission rate Mb/s Signaling – every 6th frame, last bit; signaling rate for each channel - 8 kHz/6 = kb/s Digital Communication Systems 2012 R.Sokullu 105/61

3.10. Digital Multiplexers Same concept (TDM) used for multiplexing digital signals of different rates. Conceptual diagram of multiplexing-demultiplexing. Figure 3.20 Digital Communication Systems 2012 R.Sokullu 107/61

Multiplexing is accomplished by bit-by-bit interleaving; selector switch – sequentially scanning incoming line; at the receiving side – separation into low speed components. Types of multiplexers: relatively low data bit rate user streams are TD multiplexed over the public switched telephone network. data transmission service by telecommunication carriers; part of the national digital TDM hierarchy. mmmmmmm Digital Communication Systems 2012 R.Sokullu 108/61 108

North American Digital TDM Hierarchy First level multiplexers – kb/s streams (primary rate) into a DS1 (digital signal 1) stream of Mb/s carried on the T1 system. Second level multiplexers – 4 DS1 streams into a DS2 stream at Mb/s Third level multiplexers – 7 DS2 streams into a DS3 stream at Mb/s Fourth level multiplexers – 6 DS3 into a DS4 stream at Mb/s Fifth level multiplexers – 2 DS4 streams into a DS5 at Mb/s Digital Communication Systems 2012 R.Sokullu 109/61

Important Note: Digital transmission facilities ONLY carry bit streams without interpreting what the bits themselves mean. The two sides have common understanding of how to interpret the bits: voice, data, framing format, signaling format etc. Digital Communication Systems 2012 R.Sokullu 110/61

Problems: 1. Digital signals cannot be directly interleaved into a format that allows for their separation automatically. Common clock or perfect synchronizations is needed. The multiplexed signal must include some form of framing so the individual streams can be identified at the source. The multiplexer should be able to handle small variations in bit rates – bit stuffing. Digital Communication Systems 2012 R.Sokullu 111/61

Bit stuffing To make the outgoing rate of the multiplexer a little bit higher than the sum of the max expected input rates. Each input is fed into an elastic store at the multiplexer (reading can be done at different rate). Identify stuffed bits – example AT&T M12 Multiplexer. Digital Communication Systems 2012 R.Sokullu 112/61

Example: signal format of the AT&T M12 Multiplexer Designed to combine 4 DS1 into one DS2 bit stream Each frame contains total of 24 control bits, separated by sequences of 48 data bits 4 frames, transmitted one after the other 12 bits from each input bit-by-bit interleaved, 48 bits Four types of control bits – F,M and C inserted by multiplexer – total of 24 control bits Digital Communication Systems 2012 R.Sokullu 113/61

Signal format of AT&T M12 multiplexer Figure 3.21 Digital Communication Systems 2012 R.Sokullu 114/61

Control bits F – 2 per subframe; main framing pulses ( ) M – 1 pr subframe; secondary framing, identifying the subframes (0111) C – 3 per subframe; stuffing indicators; indexes denote input channel; first subframe has 3 C bits, indicating stuffing in first DS1 stream; value 1 of all three indicates stuffed bits; value 0 – no stuffed bits; majority logic decoding if there is stuffing position of stuffing is – first bit after F1 Digital Communication Systems 2012 R.Sokullu 115/61

Receiver 1. Searches for main framing sequence – in F bits 2. Establishes the identity of the four DS1 streams and position of M and C bits 3. From the position of the M bits the correct position of the C bits is verified 4. Streams properly demultiplexed and destuffed. Safeguards: Double checking F and M bits for framing. Single error correction capability built into the C-control bits ensures that the 4 DS1 streams are properly destuffed Digital Communication Systems 2012 R.Sokullu 116/61

3.11. Virtues, Limitations and Modifications of PCM Advantages: 1. Robustness to channel noise and interference. 2. Signal regeneration possibilities along the path. 3. Efficient trade-off between increased bandwidth and improved SNR (exponential law) 4. Integration of different base-band signals. 5. Comparative easy of add and drop sources. 6. Secure communication (special modulation, encryption). Digital Communication Systems 2012 R.Sokullu 118/61

Disadvantages: 1. Increases complexity - VLSI technology 2. Increased bandwidth – satellites and fiber optic cables; data compression techniques; Digital Communication Systems 2012 R.Sokullu 119/61

Home reading assignment Conditions for optimality of Scalar Quantizers –Haykin, p.198 – 201. Provide one A4 page summary on what you have read. To be uploaded on the site. Digital Communication Systems 2012 R.Sokullu 120/61

CHAPTER 3 DELTA MODULATION
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Outline 3.12 Delta Modulation Delta Sigma Modulation 3.13 Linear Prediction 3.14 Differential Pulse Code Modulation 3.15 Adaptive Differential Pulse Code Modulation Digital Communication Systems R.Sokullu 122/45

3.12 Delta Modulation Definition: Delta Modulation is a technique which provides a staircase approximation to an over-sampled version of the message signal (analog input). sampling is at a rate higher than the Nyquist rate – aims at increasing the correlation between adjacent samples; simplifies quantizing of the encoded signal Digital Communication Systems R.Sokullu 123/45

Illustration of the delta modulation process Figure 3.22 Digital Communication Systems R.Sokullu 124/45

Principle Operation message signal is over-sampled difference between the input and the approximation is quantized in two levels - +/-Δ these levels correspond to positive/negative differences provided signal does not change very rapidly the approximation remains within +/-Δ Digital Communication Systems R.Sokullu 125/45

Assumptions and model We assume that: m(t) denotes the input message signal mq(t) denotes the staircase approximation m[n] = m(nTs), n = +/-1, +/-2 … denotes a sample of the signal m(t) at time t=nTs, where TS is the sampling period then Digital Communication Systems R.Sokullu 126/45

we can express the basic principles of the delta modulation in a mathematical form as follow:
error signal quantized error signal quantized output Digital Communication Systems R.Sokullu 127/45

Pros and cons Main advantage – simplicity Sampled version of the message is applied to a modulator (comparator, quantizer, accumulator) delay in accumulator is “unit delay” = one sample period (z-1) Digital Communication Systems R.Sokullu 128/45

Figure DM system. (a) Transmitter. (b) Receiver. Digital Communication Systems R.Sokullu 129/45

Transmitter Side comparator – computes difference between input signal and one interval delayed version of it quantizer – includes a hard- limiter with an input-output relation a scaled version of the signum function accumulator – produces the approximation mq[n] (final result) at each step by adding either +Δ or –Δ = tracking input samples by one step at a time Digital Communication Systems R.Sokullu 130/45

Receiver Side decoder – creates the sequence of positive or negative pulses accumulator – creates the staircase approximation mq[n] similar to tx side out-of-band noise is cut off by low-pass filter (bandwidth equal to original message bandwidth) Digital Communication Systems R.Sokullu 131/45

Noise in Delta Modulation Systems slope overhead distortion granular noise Digital Communication Systems R.Sokullu 132/45

Slope Overhead Distortion
The quantized message signal can be represented as: where the input to the quantizer can be represented as: Sample of m(T) at time nT Quantizer input at time (n-1)T So, (except for the quantization error) the quantizer input is the first backward difference (derivative) of the input signal = inverse of the digital integration process Digital Communication Systems R.Sokullu 133/45

Discussion Consider the max slope of the input signal m(t)
To increase the samples {mq[n]} as fast as the input signal in its max slope region the following condition should be fulfilled: otherwise the step-size Δ is too small Digital Communication Systems R.Sokullu 134/45

Granular Noise In contrast to slope overhead Occurs when step size is too large Usually relatively flat segment of the signal Analogous to quantization noise in PCM systems Digital Communication Systems R.Sokullu 135/45

Conclusion: 1. Large step-size is necessary to accommodate a wide dynamic range 2. Small step-size is required for accuracy with low-level signals = compromise between slope overhead and granular noise = adaptive delta modulation, where the step size is made to vary with the input signal (3.16) Digital Communication Systems R.Sokullu 136/45

Delta Sigma Modulation Conventional delta modulation - Quantizer input is an approximation of the derivative of the input message signal m(t). Results in the accumulation of error (noise) accumulated noise (transmission disturbances) at the receiver (cumulative error). Possible solution: integrating the message before delta modulation – called delta sigma modulation Digital Communication Systems R.Sokullu 138/45

Remark 1: The message signal is defined in its continuous form – so pulse modulator contains a hard limiter and a pulse generator to produce a 1-bit encoded signal integration at the tx requires differentiation at the rx side. But: As in conventional DM the message has to be integrated at the final stage this eliminates the need of differentiation here. Digital Communication Systems R.Sokullu 139/45

Block diagrams of systems for realizing Delta-Sigma Modulation Figure 3.25 Digital Communication Systems R.Sokullu 140/45

Remark 2: Integration is a linear operation Int 1 and Int 2 can be combined in a single integrator placed after the comparator (previous slide – 3.25 b) Results in a simpler version of DSM scheme Digital Communication Systems R.Sokullu 141/45

Pros and cons for DSM Simplicity of implementation both at the tx and rx side Requires sampling rate far in excess of the Nyquist rate (PCM) – increase in transmission and channel bandwidth If bandwidth is at a premium we have to choose increased system complexity (additional signal processing) to achieve reduced bandwidth. Digital Communication Systems R.Sokullu 142/45

How does it work? Reading assignment: 3.13 Linear Prediction (plus all that you are taught in the Signals and Systems – part II) Digital Communication Systems R.Sokullu 143/45

3.14 Differential PCM Sampling at higher then Nyquist rate creates correlation between samples (good and bad) Difference between samples has small variance – smaller than the variance of the signal itself Encoded signal contains redundant information Can be used to a positive end – remove redundancy before encoding to get a more efficient signal to be transmitted Digital Communication Systems R.Sokullu 145/45

How it works – the background
We know the signal up to a certain time Use prediction to estimate future values Signal sampled at fs= 1/Ts ; sampled sequence – {m[n]}, where samples are Ts seconds apart Input signal to the quantizer – difference between the unquantized input signal m(t) and its prediction: prediction of the input sample Digital Communication Systems R.Sokullu 146/45

Predicted value – achieved by linear prediction filter whose input is the quantized version of the input sample m[n]. The difference e[n] is the prediction error (what we expect and what actually happens) By encoding the quantizer output we actually create a variation of PCM called differential PCM (DPCM). Digital Communication Systems R.Sokullu 147/45

Figure DPCM system. (a) Transmitter. (b) Receiver. Digital Communication Systems R.Sokullu 148/45

Details: Block scheme is very similar to DM quantizer input quantizer output may be expressed as: prediction filter output may be expressed as: Digital Communication Systems R.Sokullu 149/45

If we substitute 3.75 into 3.76 we get:
sum is equal to input sample Quantized input of the prediction filter - Digital Communication Systems R.Sokullu 150/45

Details – cont’d mq[n] is the quantized version of the input sample m[n] so, irrespective of the properties of the prediction filter the quantized sample mq[n] at the prediction filter input differs from the original sample m[n] with the quantization error q[n]. If the prediction filter is good, the variance of the prediction error e[n] will be smaller than the variance of m[n] This means that if we make a very good prediction filter (adjust the number of levels) it will be possible to produce a quantization error with a smaller variance than if the input sample m[n] is quantized directly as in standard PCM Digital Communication Systems R.Sokullu 151/45

Receiver side decoder – constructs the quantized error signal quantized version of the input is recovered by using the same prediction filter as at the tx if there is no channel noise – encoded input to the decoder is identical to the transmitter output then the receiver output will be equal to mq[n] (differs from m[n] by q[n] caused by quantizing the prediction error e[n]) Digital Communication Systems R.Sokullu 152/45

Comparison DPCM and DM DPCM includes DM as a special case Similarities subject to slope-overhead and quantization error Differences DM uses a 1-bit quantizer DM uses a single delay element (zero prediction order) DPCM and PCM both DM and DPCM use feedback while PCM does not all subject to quantization error Digital Communication Systems R.Sokullu 153/45

Processing Gain Output signal-to-noise ratio (SNRO) σM2 – variance of m[n] σQ2 – variance of quantization error q[n] rewrite using variance of the prediction error σE2 processing gain signal-to-quanti-zation noise ratio Digital Communication Systems R.Sokullu 154/45

The processing gain Gp when greater than unity represents the signal-to-noise ratio that is due to the differential quantization scheme. For a given input message signal σM is fixed, so the smaller the σE the greater the Gp. This is the design objective of the prediction filter For voice signals – optimal main advantage of DPCM over PCM is b/n 4-11 dB Advantage expressed in terms of bit rate (bits) 1 bit =6 dB of quantization noise (Table 3.35, p 198) So for fixed SNR, sampling rate 8 kHz – DCPM provides saving of 8-16 kb/s (1 -2 bits per sample) PCM Digital Communication Systems R.Sokullu 155/45

3.15 Adaptive Differential PCM PCM for speech coding of 64 kb/s requires high channel bandwidth some applications (secure transmission over radio channel – low capacity) requires speech coding at low bit rates but preserving acceptable fidelity (not 64 kb/s PCM but 32, 16, 8 etc) possible using special coders that utilize statistical characteristics of speech signals and properties of hearing Digital Communication Systems R.Sokullu 157/45

Design Objectives 1. Remove redundancies from speech signals 2. Assign available bits to encode non-redundant parts of speech signal in an efficient way Standard PCM is at 64 kb/s – can be reduced to 32, 16, 8 or even 4 kb/s Price = proportionally increased complexity For same speech quality but Half the bit rate - Computational complexity is an order of magnitude higher Digital Communication Systems R.Sokullu 158/45

ADPCM principles Allows encoding of speech at 32 kb/s – requires 4 bits per sample Uses adaptive quantization and adaptive prediction adaptive quantization – uses a time-varying step Δ[n]. The step-size is varied to match the input signal σM2 φ is a constant; the other – estimate of the standard deviation - has to be computed continuously Digital Communication Systems R.Sokullu 159/45

Two possibilities: adaptive quantization with forward estimation (AQF) – uses unquantized samples of the input signal to derive forward estimates of σM[n]; requires a buffer to store samples for a certain learning period; incurs delay (~ 16 ms for speech) adaptive quantization with backward estimation (AQB) – uses samples of the quantizer output to derive backwards estimates of σM[n] Digital Communication Systems R.Sokullu 160/45

Adaptive prediction in ADPCM adaptive prediction with forward estimation (APF); uses unqunatized samples of the input signal to calculate prediction coefficients; disadvantages similar to AQF adaptive prediction with backward estimation (APB); uses samples of the quantizer output and the prediction error to compute predictor coefficients; logic for adaptive prediction – algorithm for updating predictor coefficients Digital Communication Systems R.Sokullu 161/45

Adaptive quantization with backward estimation (AQB).
Figure 3.29 Digital Communication Systems R.Sokullu 162/45

Adaptive prediction with backward estimation (APB). Figure 3.30 Digital Communication Systems R.Sokullu 163/45

Conclusion: PCM at 64 kb/s and ADPCM at 32 kb/s are internationally accepted standards for voice coding and decoding. Digital Communication Systems R.Sokullu 164/44

CHAPTER 4 BASEBAND PULSE TRANSMISSION
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Outline 4.1 Introduction 4.2. Matched Filter 4.3 Error Due to Noise 4.4 Intersymbol Interference (ISI) Digital Communication Systems R.Sokullu 166/46

4.1 Introduction In Ch.3 – methods for digital transmission of analog information bearing signals In Ch.4 – methods for digital transmission of digital information using baseband channel Digital data – broad spectrum; low-frequency components; Transmission channel bandwidth – should accommodate the essential frequency content of the data stream Channel is dispersive channel is noisy – control over additive white noise (old problem..) received signal pulses are affected by adjacent symbols (new problem) – intersymbol interference (ISI); major source of interference; Distorted pulse shape (new problem) - channel requires control over pulse shape Digital Communication Systems R.Sokullu 167/46

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Main issue to be discussed: Detection of digital pulses corrupted by the effect of the channel We know the shape of the transmitted pulse Device to be used – matched filter Digital Communication Systems R.Sokullu 169/46

4.2 Matched Filter Basic task – detecting transmitted pulses at the front end of the receiver (corrupted by noise) Receiver model Digital Communication Systems R.Sokullu 171/46

Details The filter input x(t) is: where T is an arbitrary observation interval g(t) is a binary symbol 1 or 0 w(t) is a sample function of white noise, zero mean, psd N0/2 The function of the receiver is to detect the pulse g(t) in an optimum manner, providing that the shape of the pulse is known and the distortion is due to effects of noise = To optimize the design of a filter so as to minimize the effects of noise at the filter output in some statistical sense. Digital Communication Systems R.Sokullu 172/46

Designing the filter Since we assume the filter is linear its output can be described as: where g0(t) is the recovered signal n(t) produced noise This is equal to maximizing the peak signal-to-noise ratio: instantaneous power in the output signal average output noise Digital Communication Systems R.Sokullu 173/46

we have to define the impulse response of the filter h(t) in such a way that the signal-to-noise ratio (4.3) is maximized. Digital Communication Systems R.Sokullu 174/46

Let us assume that: - G(f) - FT of the signal g(t); - H(f) – frequency response of the filter then: FT of the output signal g0(t)= H(f).G(f), or sampled at time t=T and no noise Digital Communication Systems R.Sokullu 175/46

Next step is to add the noise. What we know is that the power spectral density of white noise is: Thus the average power of the output noise n(t) is: Digital Communication Systems R.Sokullu 176/46

Substituting, So, given the function G(f), the problem is reduced to finding such an H(f) that would maximize η. Digital Communication Systems R.Sokullu 177/46

We use Schwartz inequality which states that for two complex functions, satisfying the conditions: the following is true: and equality holds iff: Digital Communication Systems R.Sokullu 178/46

In our case this inequality will have the form: and we can re-write the equation for the peak signal-to-noise ratio as: Digital Communication Systems R.Sokullu 179/46

Remarks: The right hand side of this equation does not depend on H(f). It depends only on: signal energy noise power spectral density Max value is for: Digital Communication Systems R.Sokullu 180/46

Let us denote the optimum value of H(f) by Hopt(f). To find it we use equation 4.10: The result: Except for a scaling coefficient k exp(-2πfT), the frequency response of the optimum filter is the same as the complex conjugate of the FT of the input signal. Digital Communication Systems R.Sokullu 181/46

Definition of the filter functions: In the frequency domain: knowing the input signal we can define the frequency response of the filter (in the frequency domain) as the FT of its complex conjugate. In the time domain… Digital Communication Systems R.Sokullu 182/46

Take inverse FT of Hopt(f): and keeping in mind that for real signals G*(f) = G(-f): Digital Communication Systems R.Sokullu 183/46

in the time domain it turns out that the impulse response of the filter, except for a scaling factor k, is a time-reversed and delayed function of the input signal This means it is “matched” to the input signal, that is why this type of time-invariant linear filters is known as “matched filter” NOTE: The only assumption for the channel noise was that it is stationary, white, with psd N0/2. Digital Communication Systems R.Sokullu 184/46

Properties of Matched Filters Property 1: A filter matched to a pulse signal g(t) of duration T is characterized by an impulse response that is time-reversed and delayed version of the input g(t): Time domain: hopt(t) = k . g(T-t) Frequency Domain: Hopt(f) = kG*(f)exp(- j2πfT) Digital Communication Systems R.Sokullu 185/46

Property 2: A matched filter is uniquely defined by the waveform of the pulse but for the: - time delay T - scaling factor k Digital Communication Systems R.Sokullu 186/46

Property 3: The peak signal-to-noise ratio of the matched filter depends only on the ratio of the signal energy to the power spectral density of the white noise at the filter input. using the inverse FT Digital Communication Systems R.Sokullu 187/46

The integral of the squared magnitude spectrum of a pulse signal with respect to frequency is equal to the signal energy E (Rayleigh Theorem) so substituting in the previous formula we get: After substitution of (4.14) into (4.7) we get the expression for the average output noise power as: Digital Communication Systems R.Sokullu 188/46

we finally get the following expression: and Digital Communication Systems R.Sokullu 189/46

Conclusion: From (4.20) we see that the dependence of the peak SNR on the input waveform g(t) has been completely removed by the matched filter. So, in evaluating the ability of a matched-filter receiver to overcome/remove additive white noise we see that all signals with equal energy are equally effective. We call the ratio E/N0 signal_energy-to-noise ratio (dimensionless) The matched filter is the optimum detector of a pulse of known shape in additive white noise. Digital Communication Systems R.Sokullu 190/46

4.3 Error Rate Due to Noise In this section we will derive some quantitative results for the performance of binary PCM systems, based on results for the matched filter. We consider the BER performance for a rectangular baseband pulse using and integrate and dump filter for detection. Digital Communication Systems R.Sokullu 192/46

The model: We assume binary PCM system based on polar NRZ signaling (encoding) symbols 1 and 0 are equal amplitude and equal duration channel noise is modeled as AWGN w(t) with zero mean and psd N0/2 the received signal, in the interval 0 ≤ t ≤ Tb can be represented as: white noise amplitude Digital Communication Systems R.Sokullu 193/46

also it is assumed that the receiver has accurate knowledge of the starting and ending times of each pulse (perfect synchronization); the receiver has to make a decision whether the pulse is a 1 or a 0. the structure of the receiver is as follows: Digital Communication Systems R.Sokullu 194/46

Details: The filter is matched to a rectangular pulse of amplitude A and duration Tb. The resulting matched filter output is sampled at the end of each signaling interval. The channel noise adds randomness to the matched filter output. Digital Communication Systems R.Sokullu 195/46

For the channel model we are discussing (AWGN), integrating the noise over a period T is equal to creating noise with Gaussian distribution with variance σ02 = N0 T/2, where N0 is the noise power in Watts/Hz. Digital Communication Systems R.Sokullu 196/46

If we consider a single symbol s(t) of voltage V passing through the detector with additive noise n(t) then the output of the integrator y(t) will be: symbol contribution noise contribution Digital Communication Systems R.Sokullu 197/46

Then the probability density of the integrated noise at the sampling point is: A detection error will occur if the noise sample exceeds - VT/2. The probability for such an event is calculated from (1) Digital Communication Systems R.Sokullu 198/46

substituting we can write the expression for the probability of error as: complementary error function Digital Communication Systems R.Sokullu 199/46

The probability of error can also be expressed in terms of the signal energy as follows: As 0 and 1 are equiprobable the result for receiving a logic 0 in error will be the same only in this case the noise sample will be exceeding +VT/2. Digital Communication Systems R.Sokullu 200/46

For the special case we are discussing unipolar NRZ the average symbol energy is half that of the logic 1. So the symbol error probability will be: Digital Communication Systems R.Sokullu 201/46

Figure 4.5 Noise analysis of PCM system. (a) Probability density function of random variable Y at matched filter output when 0 is transmitted. (b) Probability density function of Y when 1 is transmitted. Digital Communication Systems R.Sokullu 202/46

Unipolar vs Bipolar Symbols For unipolar symbols as we said the average energy is equal to E0/2. For bipolar symbols (logic 1 is conveyed by +V, logic 0 – by –V volts) it can be proved that (An example of this is RS-232, where "one" is −5V to −12V and "zero" is +5 to +12V) Digital Communication Systems R.Sokullu 203/46

4.4 PAM and Intersymbol Interference (ISI) ISI is due to the fact that the communication channel is dispersive – some frequencies of the received pulse are delayed which causes pulse distortion (change in shape and delay). Most efficient method for baseband transmission – both in terms of power and bandwidth - is PAM. Digital Communication Systems R.Sokullu 205/46

Considered model: baseband binary PAM system incoming binary sequence b0 consists of 1 and 0 symbols of duration Tb Figure 4.7 Baseband binary data transmission system. Digital Communication Systems R.Sokullu 206/46

PAM changes this sequence into a new sequence of short pulses each with amplitude ak, represented in polar form as: applied to a transmit filter of impulse response g(t): transmitted signal Digital Communication Systems R.Sokullu 207/46

s(t) is modified by a channel with channel impulse response h(t) random white noise is added (AWGN channel model) x(t) is the channel output, the noisy signal arriving at the receiver front end receiver has a receive filter with impulse response c(t) and output y(t) y(t) is sampled synchronously with the transmitter (clock signal is extracted from the receive filter output) reconstructed samples are compared to a threshold decision is taken as for 1 or 0 Digital Communication Systems R.Sokullu 208/46

Receive filter output is: should have a constant delay t0 here set to 0 The scaled pulse µp(t) can be expressed as double convolution: the impulse response of the transmit filter g(t), the impulse response of the channel filter h(t) (channel) and the impulse response of the receive filter c(t): Digital Communication Systems R.Sokullu 209/46

In the frequency domain we have:
where P(f), G(f), H(f) and C(f) are FT of the respective p(t), g(t), h(t) and c(t) The received signal is sampled at times ti= iTb which, taking (4.44) in mind and the norm. condition p(0) = 1 (µ is a scaling factor to account for amplitude changes), can be expressed as: residual effect due to other transmitted pulses contribution of the ith pulse Digital Communication Systems R.Sokullu 210/46

Conclusion: In the absence of noise and ISI we get and ideal pulse (from (4.48) ) y(ti)=µai The ISI can be controlled (reduced) by the proper design of the transmit and receive filter Subject discussed further in the following sections…. Digital Communication Systems R.Sokullu 211/47

CHAPTER 4 BASEBAND DATA TRANSMISSION

Outline 4.5 Nyquist Criterion for Distortion-less Baseband Binary Channel 4.6 Correlative-Level Coding 4.7 Baseband M-ary PAM Transmission 4.8 Digital Subscriber Lines 4.9 Optimum Linear Receiver Digital Communication Systems 2012 R.Sokullu 213/52

From the previous class.. We explained what is a “matched filter” The matched filter is the optimum detector of a pulse of known shape in additive white noise. In evaluating the ability of a matched-filter receiver to overcome/remove additive white noise we see that all signals with equal energy are equally effective. The matched filter completely removes the dependence of the peak SNR on the input waveform g(t). But still, there is the ISI.. What do we do with it from design point of view? This is the subject of today’s lecture… Digital Communication Systems 2012 R.Sokullu 214/52

We defined ISI as: Interference due to the dispersiveness – (some frequencies of the received pulse are delayed) of the communication channel which causes pulse distortion (change in shape and delay). Also we defined the transmitted and received signal in the form of a sequence of pulses as: transmitted signal received signal Digital Communication Systems 2012 R.Sokullu 215/52

taking in mind that the received signal is sampled at times ti= iTb and using (4.46) and (4.47) the received signal can be expressed as: Noise sample at time ti contribution of the ith pulse residual effect due to other transmitted pulses Digital Communication Systems 2012 R.Sokullu 216/52

4.5 Nyquist Criterion for Distortion-Less Baseband Binary Transmission This is the diagram of the binary transmission system From design point of view – frequency response of the channel and transmitted pulse shape are specified; the frequency response of the transmit and receive filters has to be determined so as to reconstruct [bk]. Done by extracting and decoding the corresponding sequence of coefficients [ak] from the output y(t). Digital Communication Systems 2012 R.Sokullu 217/52

Reconstructing: extracting sampling of the output y(t) at times t = iTb decoding the weighted pulse contribution should be free from ISI to take a proper logical decision the weighted pulse contribution is: To be free from ISI it has to meet the condition: p(0)=1 by normalization Digital Communication Systems 2012 R.Sokullu 218/52

if the condition given in (4.49) is satisfied then the pulse will be free from ISI and its form will be a perfect pulse (of course not considering the noise): So, the condition (4.49), formulated in the time domain, ensures perfect symbol recovery if there is no noise. In the next slides we will try to formulate this condition in the frequency domain. y(ti)=µai Digital Communication Systems 2012 R.Sokullu 219/52

In the Frequency Domain: If we consider a sequence of samples {p[nTb]}, where n = 0, +/-1, +/-2… in the time domain, we will have periodicity in the frequency domain (duality property) (discusses in detail in Ch.3) which in general can be expressed as: FT of an infinite sequence of samples Rb=1/Tb bit rate (b/s) Digital Communication Systems 2012 R.Sokullu 220/52

Pδ(f) in our case is the FT of an infinite sequence of delta functions with period Tb, weighted by the sample values of p(t). So, it can also be expressed by: where if we let m = i – k for i = k we have m = 0 and for i ≠ k we have correspondingly m ≠0 Digital Communication Systems 2012 R.Sokullu 221/52

further on if we impose the conditions (4.49) on the sample values of p(t): ..and using the sifting property of the delta function (4.51) can be written as: Digital Communication Systems 2012 R.Sokullu 222/52

As we have the normalizing condition p(0) = 1 substituting (4.52) in (4.50) we finally get: which represents the condition for zero ISI in the frequency domain Digital Communication Systems 2012 R.Sokullu 223/52

Conclusion: The case when ISI is equal to zero is known as distortion-less channel. We have derived the condition for distortion- less channel both in the time (4.49) and frequency domain (4.53) in the absence of noise We can formulate the Nyquist criterion for distortion-less bandpass transmission in the absence of noise as follows: Digital Communication Systems 2012 R.Sokullu 224/52

The frequency function P(f) eliminates ISI for samples taken at intervals Tb providing that it satisfies equation (4.53). It is important to note that P(f) refers to the whole system, including the transmission filter, the channel and the receiver filter in accordance with equation (4.47): Digital Communication Systems 2012 R.Sokullu 225/52

As it is very unlikely in real life that a channel itself will exhibit Nyquist transfer response and the condition for distortion-less transmission is incorporated in the design of the filters used. This is also known as Nyquist channel filtering and Nyquist channel reponse. Often the Nyquist filtering response needed for zero ISI is split between Tx and Rx using a root raised cosine filter pair (which we will discuss later) Next we discuss what is understood by “Ideal Nyquist channel” Digital Communication Systems 2012 R.Sokullu 226/52

Ideal Nyquist Channel If we have to design a filter to meet the condition in (4.53) for recovering pulses free from ISI one possible function that we can specify for the frequency function P(f) is the rectangular function: overall system bandwidth rect function of unit amplitude at f = 0 Digital Communication Systems 2012 R.Sokullu 227/52

Figure 4.8 (a) Ideal magnitude response. (b) Ideal basic pulse shape. Digital Communication Systems 2012 R.Sokullu 228/52

in the time domain this corresponds to: the special rate Rb is the well known Nyquist rate the W = Rb/2 is the Nyquist bandwidth Digital Communication Systems 2012 R.Sokullu 229/52

Conclusions: The ideal baseband pulse transmission system (channel) which satisfies eq. (4.54) in the frequency domain, eq. (4.56) in the time domain is known as the Ideal Nyquist Channel. The function p(t) is regarded as the impulse response of that channel. It is in fact the impulse response of an ideal low- pass filter with pass-band magnitude response 1/2W and bandwidth W. It has its peak at origin, and goes through zero at integer multiples of the bit duration Tb. If such a waveform is sampled at t = 0, +/-Tb, +/-2Tb.. the pulses defined by µp(t-iTb) with amplitude µ will not interfere with each other. Digital Communication Systems 2012 R.Sokullu 230/52

Figure 4.9 A series of sinc pulses corresponding to the sequence Digital Communication Systems 2012 R.Sokullu 231/52

Unfortunately, This is difficult to realize in practice because: requires flat magnitude characteristic P(f) from –W to +W, and 0 elsewhere there is no margin of error for the sampling in the receiver (p(t) decreases slowly and decays for large t) A practical solution is the raised cosine filter mentioned before. Digital Communication Systems 2012 R.Sokullu 232/52

Raised Cosine Spectrum We extend the min value of W = Rb/2 to an adjustable value between W and 2W. We specify a condition for the overall frequency response P(f) . Specifically in the equation for the ideal frequency response (4.53) we consider only three terms (three harmonics) and restrict the bandwidth to (-W, + W). Digital Communication Systems 2012 R.Sokullu 233/52

Then (4.43) reduces to the following expression: There are several possible band-limited functions to satisfy this equation. Of great practical interest is the raised cosine spectrum whose frequency domain characteristic is given on the next slide: Digital Communication Systems 2012 R.Sokullu 234/52

where f1 and the band-width W are related by indicates the excess bandwidth over the ideal solution W rolloff factor BT=2W-f1 =W(1+α) Digital Communication Systems 2012 R.Sokullu 235/52

Figure Responses for different roll-off factors. (a) Frequency response. (b) Time response. Digital Communication Systems 2012 R.Sokullu 236/52

Remarks: In Figure 4.10a we have the normalized P(f) . Increasing the roll off factor we see that it gradually cuts off compared to Ideal Niquist Channel (α = 0) In Figure 4.10b we have p(t) which is obtained from P(f) (4.60) using the FT. Digital Communication Systems 2012 R.Sokullu 237/52

factor characterizing the Ideal Nyquist Channel factor decreasing with time, proportional to 1/ |t2| ensures zero crossings of p(t) at the desired time instants t = iT reduces the tails of the pulse considerably below that of the INC Digital Communication Systems 2012 R.Sokullu 238/52

Conclusions: For α = 1 we have the most gradual cut offs and also the smallest amplitude of the tails in the time domain. This may be interpreted as the intersymbol interference resulting from timing error decreasing as the roll off factor α is increased from 0 to 1. The special case of α = 1 is known as the full-cosine roll off characteristic Its frequency response is given as: Digital Communication Systems 2012 R.Sokullu 239/52

correspondingly in the time domain given as: The time response has two interesting properties: At t = ±Tb/2 = ±1/4W p(t) = 0.5 – the pulse width measured at half amplitude is equal to the bit duration There are zero crossings at t = ±3Tb/2, ±5Tb/2, … in addition to the usual zero crossings at t = ±Tb, ±2Tb…. Note: These two properties are extremely important when extracting a timing signal from the received signal for synchronization. Price: Double bandwidth compared to the INC. Digital Communication Systems 2012 R.Sokullu 240/52

Outline 4.5 Nyquist Criterion for Distortion-less Baseband Binary Channel 4.6 Correlative-Level Coding 4.7 Baseband M-ary PAM Transmission 4.8 Digital Subscriber Lines Digital Communication Systems 2012 R.Sokullu 241/52

Introduction Uncontrolled ISI is a problem and is undesired. If it is added in a controlled manner it is possible to achieve a signaling rate equal to the Nyquist rate of 2W in a channel with bandwidth W Hertz. (theoretical maximum) Such methods are called correlative-level coding or partial-response signaling schemes. The idea is: as the ISI included in the transmitted signal is known it can be interpreted at the receiver in a deterministic way. Digital Communication Systems 2012 R.Sokullu 242/52

Duobinary Signaling Duo – example of doubling the transmission capacity (class I partial response) Assumptions: binary input sequence [bk] of uncorrelated symbols 1 and 0, each with duration Tb sequence is applied to a PAM producing a two-level sequence of short pulses (approximating unit impulse) amplitude of pulses is defined as: Digital Communication Systems 2012 R.Sokullu 243/52

sequence is applied to a duo-binary encoder (represented by filter HI(f)) and is converted to a three-level output (in our example ( -2, 0,+2) for every unit impulse we get two impulses spaced at Tb Figure 4.11 Duo-binary signaling scheme (encoder, channel, receiver). Digital Communication Systems 2012 R.Sokullu 244/52

So, the output of the duo- binary encoder ck can be expressed as follows: [ak] is uncorrelated but [ck] is correlated sequence. From a two-level binary uncorrelated sequence the duo-binary encoder has created a sequence with three levels which is correlated The correlation between adjacent pulses is actually ISI introduced in a deterministic manner Digital Communication Systems 2012 R.Sokullu 245/52

Description in the Frequency Domain An ideal delay element, with delay Tb has a frequency response of exp(-j2πfTb). The frequency response of the simple delay-line filter in fig can be given as 1 + exp(-j2πfTb) Then the overall frequency response of the filter in cascade with INC (???) is: Digital Communication Systems 2012 R.Sokullu 246/52

For an INC we have: So, the overall frequency response can be written as: and the next slide shows the magnitude and the phase response. Digital Communication Systems 2012 R.Sokullu 247/52

Figure 4.12 Frequency response of the duo-binary conversion filter. (a) Magnitude response. (b) Phase response. Digital Communication Systems 2012 R.Sokullu 248/52

Description in the Time Domain from (4.67) we can find the impulse response of HI(f) as: two sinc pulses displaced by Tb Digital Communication Systems 2012 R.Sokullu 249/52

Figure 4.13 Impulse response of the duo-binary conversion filter. Digital Communication Systems 2012 R.Sokullu 250/52

Remarks: consists of two Nyquist pulses (sinc pulses) which are time-displaced by Tb the impulse response h(t) has only two distinguishable values at the sampling instants. The response to one input pulse is spread over more than one signaling intervals so in any one interval the response is only “partial” hence the name ‘partial-response signaling” Digital Communication Systems 2012 R.Sokullu 251/52

Detection of Duobinary Signals Based on equation (4.66): Let denote the estimate of the original pulse ak as conceived by the receiver at time t = kt Then Digital Communication Systems 2012 R.Sokullu 252/52

So, if ck is received without an error, and the previous estimate was right, then the current estimate of ak will also be correct. this technique is using stored information from the previous symbol – known as decision feedback. But, once error is made it propagates forward, because each new decision depends on the previous. To avoid errors: using precoding (EXCLUSIVE OR) Digital Communication Systems 2012 R.Sokullu 253/52

Figure 4.14 A precoded duobinary scheme; details of the duobinary coder are given in Figure 4.11. Digital Communication Systems 2012 R.Sokullu 254/52

The pre-coded binary sequence [dk] is applied to a PAM producing the corresponding two level sequence of short pulses [ak]. (-1 or +1) Duo-binary encoding is linear, but pre-coding is not. When we put (4.72) and (4.74) together we get: Digital Communication Systems 2012 R.Sokullu 255/52

so the decision is taken as: Figure 4.15 Detector for recovering original binary sequence from the pre-coded duo-binary coder output. Digital Communication Systems 2012 R.Sokullu 256/52

In this model the PAM block produces binary pulses (two possible amplitude levels) A PAM in a baseband M-ary PAM system the pulse amplitude modulator produces one of M possible amplitude levels (M>2). Next slide – example for quaternary (M=4) – each level represents a dibit (pairs of bits) Digital Communication Systems 2012 R.Sokullu 258/52

Figure 4.20 Output of a quaternary system. (a) Waveform. (b) Representation of the 4 possible dibits, based on Gray encoding. Digital Communication Systems 2012 R.Sokullu 259/52

In an M-ary system we have the following relationships: more than one bit is encoded in a symbol forming an alphabet if symbols differ with only 1 bit we have Gray Coding each symbol is represented by one amplitude level each symbol duration Tb; all symbols equally likely and statistically independent Then: symbol rate is 1/Tb – rate of emitting symbols per second (baud rate) for binary - bit rate is 1/Tb – rate of sending bits for quaternary – 4 possible bits; sent in combinations of two – so: 1 baud = 2 bits per second Generalized: 1 baud = log2M bits per second or Digital Communication Systems 2012 R.Sokullu 260/52

Remarks: In a given channel bandwidth an M-ary system can transmit bits at a rate log2M faster than the corresponding binary system. To realize this the M-ary system requires more transmission power – average of M2/log2M compared to binary. Algorithm of an M-ary system Source symbols converted into an M-ary PAM pulse train (PAM block) Pulse train is shaped by a transmit filter Transmit over a channel (AWGN) – noise and distortion. Passed trough a received filter. Sampled at appropriate rate in sync with the transmitter Each sample compared to a threshold – decision is taken which symbol was transmitted Same as binary PAM – ISI, noise, timing errors; same procedures for designing transmit and receive filter but more complex. Digital Communication Systems 2012 R.Sokullu 261/52

Reading Assignments Reading assignment 2: 4.8 Digital Subscriber Lines – pp. 277 – 282 Reading Assignment 3: 4.9 Optimum Linear Receiver – pp. 282 – 287. Digital Communication Systems 2012 R.Sokullu 263/52

CHAPTER 5 SIGNAL SPACE ANALYSIS

Outline 5.1 Introduction 5.2 Geometric Representation of Signals Gram-Schmidt Orthogonalization Procedure 5.3 Conversion of the AWGN into a Vector Channel 5.4 Maximum Likelihood Decoding 5.5 Correlation Receiver 5.6 Probability of Error Digital Communication Systems 2012 R.Sokullu 265/45

Introduction – the Model We consider the following model of a generic transmission system (digital source): A message source transmits 1 symbol every T sec Symbols belong to an alphabet M (m1, m2, …mM) Binary – symbols are 0s and 1s Quaternary PCM – symbols are 00, 01, 10, 11 Digital Communication Systems 2012 R.Sokullu 266/45

Transmitter Side Symbol generation (message) is probabilistic, with a priori probabilities p1, p2, .. pM. or Symbols are equally likely So, probability that symbol mi will be emitted: Digital Communication Systems 2012 R.Sokullu 267/45

Transmitter takes the symbol (data) mi (digital message source output) and encodes it into a distinct signal si(t). The signal si(t) occupies the whole slot T allotted to symbol mi. si(t) is a real valued energy signal (???) Digital Communication Systems 2012 R.Sokullu 268/45

Transmitter takes the symbol (data) mi (digital message source output) and encodes it into a distinct signal si(t). The signal si(t) occupies the whole slot T allotted to symbol mi. si(t) is a real valued energy signal (signal with finite energy) Digital Communication Systems 2012 R.Sokullu 269/45

Channel Assumptions: Linear, wide enough to accommodate the signal si(t) with no or negligible distortion Channel noise is w(t) is a zero-mean white Gaussian noise process – AWGN additive noise received signal may be expressed as: Digital Communication Systems 2012 R.Sokullu 270/45

Receiver Side Observes the received signal x(t) for a duration of time T sec Makes an estimate of the transmitted signal si(t) (eq. symbol mi). Process is statistical presence of noise errors So, receiver has to be designed for minimizing the average probability of error (Pe) What is this? Pe = Symbol sent cond. error probability given ith symbol was sent Digital Communication Systems 2012 R.Sokullu 271/45

5.2. Geometric Representation of Signals
Objective: To represent any set of M energy signals {si(t)} as linear combinations of N orthogonal basis functions, where N ≤ M Real value energy signals s1(t), s2(t),..sM(t), each of duration T sec Orthogonal basis function coefficient Energy signal Digital Communication Systems 2012 R.Sokullu 273/45

Coefficients: Real-valued basis functions: Digital Communication Systems 2012 R.Sokullu 274/45

The set of coefficients can be viewed as a N- dimensional vector, denoted by si Bears a one-to-one relationship with the transmitted signal si(t) Digital Communication Systems 2012 R.Sokullu 275/45

Figure 5.3 (a) Synthesizer for generating the signal si(t). (b) Analyzer for generating the set of signal vectors si. Digital Communication Systems 2012 R.Sokullu 276/45

Each signal in the set si(t) is completely determined by the vector of its coefficients Digital Communication Systems 2012 R.Sokullu 277/45

Finally, The signal vector si concept can be extended to 2D, 3D etc. N- dimensional Euclidian space Provides mathematical basis for the geometric representation of energy signals that is used in noise analysis Allows definition of Length of vectors (absolute value) Angles between vectors Squared value (inner product of si with itself) Matrix Transposition Digital Communication Systems 2012 R.Sokullu 278/45

Figure 5.4 Illustrating the geometric representation of signals for the case when N  2 and M  3. (two dimensional space, three signals) Digital Communication Systems 2012 R.Sokullu 279/45

Also, What is the relation between the vector representation of a signal and its energy value? …start with the definition of average energy in a signal…(5.10) Where si(t) is as in (5.5): Digital Communication Systems 2012 R.Sokullu 280/45

Φj(t) is orthogonal, so finally we have:
After substitution: After regrouping: Φj(t) is orthogonal, so finally we have: The energy of a signal is equal to the squared length of its vector Digital Communication Systems 2012 R.Sokullu 281/45

Formulas for two signals
Assume we have a pair of signals: si(t) and sj(t), each represented by its vector, Then: Inner product is invariant to the selection of basis functions Inner product of the signals is equal to the inner product of their vector representations [0,T] Digital Communication Systems 2012 R.Sokullu 282/45

Euclidian Distance The Euclidean distance between two points represented by vectors (signal vectors) is equal to ||si-sk|| and the squared value is given by: Digital Communication Systems 2012 R.Sokullu 283/45

Angle between two signals The cosine of the angle Θik between two signal vectors si and sk is equal to the inner product of these two vectors, divided by the product of their norms: So the two signal vectors are orthogonal if their inner product siTsk is zero (cos Θik = 0) Digital Communication Systems 2012 R.Sokullu 284/45

Schwartz Inequality Defined as: accept without proof… Digital Communication Systems 2012 R.Sokullu 285/45

Gram-Schmidt Orthogonalization Procedure Assume a set of M energy signals denoted by s1(t), s2(t), .. , sM(t). Define the first basis function starting with s1 as: (where E is the energy of the signal) (based on 5.12) Then express s1(t) using the basis function and an energy related coefficient s11 as: Later using s2 define the coefficient s21 as: Digital Communication Systems 2012 R.Sokullu 287/45

If we introduce the intermediate function g2 as: We can define the second basis function φ2(t) as: Which after substitution of g2(t) using s1(t) and s2(t) it becomes: Note that φ1(t) and φ2(t) are orthogonal that means: Orthogonal to φ1(t) (Look at 5.23) Digital Communication Systems 2012 R.Sokullu 288/45

And so on for N dimensional space…, In general a basis function can be defined using the following formula: where the coefficients can be defined using: Digital Communication Systems 2012 R.Sokullu 289/45

Special case: For the special case of i = 1 gi(t) reduces to si(t). General case: Given a function gi(t) we can define a set of basis functions, which form an orthogonal set, as: Digital Communication Systems 2012 R.Sokullu 290/45

Conversion of the Continuous AWGN Channel into a Vector Channel Suppose that the si(t) is not any signal, but specifically the signal at the receiver side, defined in accordance with an AWGN channel: So the output of the correlator (Fig. 5.3b) can be defined as: Digital Communication Systems 2012 R.Sokullu 292/45

deterministic quantity random quantity contributed by the transmitted signal si(t) sample value of the variable Wi due to noise Digital Communication Systems 2012 R.Sokullu 293/45

Now, Consider a random process X1(t), with x1(t), a sample function which is related to the received signal x(t) as follows: Using 5.28, and and the expansion 5.5 we get: which means that the sample function x1(t) depends only on the channel noise! Digital Communication Systems 2012 R.Sokullu 294/45

The received signal can be expressed as: NOTE: This is an expansion similar to the one in 5.5 but it is random, due to the additive noise. Digital Communication Systems 2012 R.Sokullu 295/45

Statistical Characterization The received signal (output of the correlator of Fig.5.3b) is a random signal. To describe it we need to use statistical methods – mean and variance. The assumptions are: X(t) denotes a random process, a sample function of which is represented by the received signal x(t). Xj(t) denotes a random variable whose sample value is represented by the correlator output xj(t), j = 1, 2, …N. We have assumed AWGN, so the noise is Gaussian, so X(t) is a Gaussian process and being a Gaussian RV, X j is described fully by its mean value and variance. Digital Communication Systems 2012 R.Sokullu 296/45

Mean Value Let Wj, denote a random variable, represented by its sample value wj, produced by the jth correlator in response to the Gaussian noise component w(t). So it has zero mean (by definition of the AWGN model) …then the mean of Xj depends only on sij: Digital Communication Systems 2012 R.Sokullu 297/45

Variance Starting from the definition, we substitute using 5.29 and 5.31 Autocorrelation function of the noise process Digital Communication Systems 2012 R.Sokullu 298/45

It can be expressed as: (because the noise is stationary and with a constant power spectral density) After substitution for the variance we get: And since φj(t) has unit energy for the variance we finally have: Correlator outputs, denoted by Xj have variance equal to the power spectral density N0/2 of the noise process W(t). Digital Communication Systems 2012 R.Sokullu 299/45

Properties (without proof) Xj are mutually uncorrelated Xj are statistically independent (follows from above because Xj are Gaussian) and for a memoryless channel the following equation is true: Digital Communication Systems 2012 R.Sokullu 300/45

Define (construct) a vector X of N random variables, X1, X2, …XN, whose elements are independent Gaussian RV with mean values sij, (output of the correlator, deterministic part of the signal defined by the signal transmitted) and variance equal to N0/2 (output of the correlator, random part, calculated noise added by the channel). then the X1, X2, …XN , elements of X are statistically independent. So, we can express the conditional probability of X, given si(t) (correspondingly symbol mi) as a product of the conditional density functions (fx) of its individual elements fxj. NOTE: This is equal to finding an expression of the probability of a received symbol given a specific symbol was sent, assuming a memoryless channel Digital Communication Systems 2012 R.Sokullu 301/45

…that is: where, the vector x and the scalar xj, are sample values of the random vector X and the random variable Xj. Digital Communication Systems 2012 R.Sokullu 302/45

Vector x is called observation vector Scalar xj is called observable element Vector x and scalar xj are sample values of the random vector X and the random variable Xj Digital Communication Systems 2012 R.Sokullu 303/45

Since, each Xj is Gaussian with mean sj and variance N0/2 we can substitute in 5.44 to get 5.46: Digital Communication Systems 2012 R.Sokullu 304/45

If we go back to the formulation of the received signal through a AWGN channel 5.34 Only projections of the noise onto the basis functions of the signal set {si(t)Mi=1 affect the significant statistics of the detection problem The vector that we have constructed fully defines this part Digital Communication Systems 2012 R.Sokullu 305/45

Finally, The AWGN channel, is equivalent to an N- dimensional vector channel, described by the observation vector Digital Communication Systems 2012 R.Sokullu 306/45

Maximum Likelihood Decoding to be continued…. Digital Communication Systems 2012 R.Sokullu 308/45

CHAPTER 5 SIGNAL SPACE ANALYSIS
Digital Communication Systems 2012 R.Sokullu 309/26 309

Outline 5.1 Introduction 5.2 Geometric Representation of Signals Gram-Schmidt Orthogonalization Procedure 5.3 Conversion of the AWGN into a Vector Channel 5.4 Likelihood Functions 5.5 Maximum Likelihood Decoding 5.6 Correlation Receiver 5.7 Probability of Error Digital Communication Systems 2012 R.Sokullu 310/26 310

Likelihood Functions In a sense, likelihood works backwards from probability: given parameter B, we use the conditional probability P(A|B) to reason about outcome A, and given outcome A, we use the likelihood function L(B|A) to reason about parameter B. This mode of reasoning is formalized in Bayes' theorem: A likelihood function is a conditional probability function considered as a function of its second argument with its first argument held fixed, thus: and also any other function proportional to such a function. That is, the likelihood function for B is the equivalence class of functions Digital Communication Systems 2012 R.Sokullu 311/26 311

Likelihood Functions As we discussed in the previous class, the conditional probability density functions fX(x|mi), I = 1, 2, 3, …M are the very characterization of the AWGN channel. They express the functional dependence of the observation vector x on the transmitted message symbol mi. (known as the transmitted message symbol) Digital Communication Systems 2012 R.Sokullu 312/26 312

However, If we have the observation vector given, and we want to define the transmitted message signal, then we have the reverse situation We introduce the “likelihood function” L(mi) as: Yes, but meaning is different… Looks very similar???? Or log likelihood function ..l(mi) as: Digital Communication Systems 2012 R.Sokullu 313/26 313

Log-Likelihood Function of AWGN Channel
Substitute 5.46 into 5.50: Vector presentation of the AWGN channel where sij, j = 1, 2, 3, ..N are the elements of the signal vector si, representing the message symbol mi. Digital Communication Systems 2012 R.Sokullu 314/26 314

So, which is the log likelihood function of the AWGN channel.. Digital Communication Systems 2012 R.Sokullu 315/26 315

5.5 Maximum Likelihood Decoding Defining the problem Suppose that in each time slot duration of T seconds, one of M possible signals, s1(t), s2(t), …sM(t) is transmitted with equal probability, 1/M. As described in the previous part, for the vector representation, the signal si(t), i=1, 2, …M is applied to a bank of correlators, with a common input and supplied with a suitable set of N orthogonal basis functions, N. The resulting output defines the signal vector si. We represent each signal si(t) as a point in the Euclidian space, N ≤ M (referred to as transmitted signal point or message point).The set of message points corresponding to the set of transmitted signals si(t) {i = 1 to M} is called signal constellation. Digital Communication Systems 2012 R.Sokullu 317/26 317

Figure 5.3 (a) Synthesizer for generating the signal si(t). (b) Analyzer for generating the set of signal vectors si. Digital Communication Systems 2012 R.Sokullu 318/26 318

The received signal x(t) is applied to a bank of N correlators (Fig. 5.3b) and the correlator outputs define the observation vector x. On the receiving side the representation of the received signal x(t) is complicated by the additive noise w(t). As we discussed the previous class, the vector x differs from the vector si by the noise vector w. However only the portion of it which interferes with the detection process is of importance to us, and this is fully described by w(t). Digital Communication Systems 2012 R.Sokullu 319/26 319

Based on the observation vector x we may represent the received signal signal x(t) by a point in the same Euclidian space used to represent the transmitted signal. Digital Communication Systems 2012 R.Sokullu 320/26 320

For a given observation vector x we have to make a decision m' = mi The decision is based on the criterion to minimize the probability of error in mapping each observation vector into a decision. So the optimum decision rule is: Digital Communication Systems 2012 R.Sokullu 321/26 321

The same rule can be more explicitly expressed using the a priori probabilities of the transmitted signals as: Conditional pdf of observation vector X given mk was transmitted Unconditional pdf of observation vector X a priori probability of transmitting mk Digital Communication Systems 2012 R.Sokullu 322/26 322

Thus we can conclude, according to the definition of likelihood functions, the likelihood function l(mk) will be maximum for k = i. So the decision rule using the likelihood function will be formulated as: For a graphical representation of the maximum likelihood rule we introduce the following: Observation space – Z, N-dimensional, consisting of all possible observation vectors x Z is partitioned into M decision regions, Z1, Z2, .. ZM Digital Communication Systems 2012 R.Sokullu 323/26 323

For the AWGN channel.. Based on the log-likelihood function, of the AWGN channel, l(mk) will be max when the term: is minimized by k = i. Decision rule for AWGN: Or using Euclidian space notation Digital Communication Systems 2012 R.Sokullu 324/26 324

Finally, (5.59) states that the maximum likelihood decision rule is simply to choose the message point closest to the received signal point. After few re-organizations we get: (left as homework brain gymnastic exercise for you) Energy of the transmitted signal sk(t) Digital Communication Systems 2012 R.Sokullu 325/26 325

Figure 5.8 Illustrating the partitioning of the observation space into decision regions for the case when N  2 and M  4; it is assumed that the M transmitted symbols are equally likely. Digital Communication Systems 2012 R.Sokullu 326/26 326

5.6 Correlation Receiver Based on the theoretical assumptions made in the previous class we define the correlator at the receiver side. It can be implemented as a optimum receiver that consists of two parts: Detector part – M product-integrators supplied with the corresponding set of coherent reference signals (orthogonal basis functions), generated locally. It operates on the received signal s(t) to produce the observation vector x for 0≤ t ≤ T. Receiver part – signal transmission decoder – which is implemented in the form of a maximum likelihood decoder, operating on the observation vector x to produce the estimate m‘ of the transmitted symbol mi in a way to minimize the average probability of symbol error. According to (5.61) the N elements of the observation vector x are multiplied by the N elements of each of the M signal vectors s1, s2, ..sM and then summed up to produce the inner products [xTsk|k=1,2..M]. Largest of the resulting numbers is selected. Digital Communication Systems 2012 R.Sokullu 328/26 328

Figure 5.9 (a) Detector or demodulator (b) Signal transmission decoder. Digital Communication Systems 2012 R.Sokullu 329/26 329

Note: The detector shown in Fig. 5.9a is based on correlators. Alternatively, matched filters, discussed in Chap may be used to produce the required observation vector x. Detector part of matched filter receiver; the signal transmission decoder is as shown in Fig. 5.9b Digital Communication Systems 2012 R.Sokullu 330/26 330

5.7 Probability of Error To complete the statistical characterization of the correlation receiver (Fig. 5.9) we need to discuss its noise performance. Using the assumptions made before, we can define the average probability of error Pe as: Digital Communication Systems 2012 R.Sokullu 332/26 332

Using the likelihood function this can be re-written as: The probability of error is invariant to rotation and translation of the signal constellation. In maximum likelihood detection the probability of symbol error Pe depends solely on the Euclidian distances between the message points in the constellation The additive Gaussian noise is spherically symmetric in all directions in the signal space. Digital Communication Systems 2012 R.Sokullu 333/26 333

Conclusions: This chapter presents a systematic procedure for the analysis of signals in a vector space. The basic idea of the approach is to represent each member of a set of transmitted signals by an N-dimensional vector, where N is the number of orthogonal basis functions, needed for the unique representation of the transmitted signals. The set of signal vectors defines the signal constellation, the N-dimensional space defines the signal space. It is the theoretical basis for the design of a digital communication receiver in the presence of AWGN. The procedure is based on the theory of maximum likelihood detection. The average probability of symbol error is defined as Pe. It is dominated by the nearest neighbors to the transmitted signal. Digital Communication Systems 2012 R.Sokullu 334/26 334

CHAPTER 6 PASS-BAND DATA TRANSMISSION

Outline 6.1. Introduction 6.2. Pass-band Transmission 6.3 Coherent Phase Shift Keying - BPSK Digital Communication Systems 2012 R.Sokullu 336/62

6.1 Introduction In Ch. 4 we studied digital baseband transmission where the generated data stream, represented in the form of discrete pulse-amplitude modulated signal (PAM) is transmitted directly over a low-pass channel. In Ch.6 we will study digital pass-band transmission where the incoming digital signal is modulated onto a carrier (usually sinusoidal) with fixed frequency limits imposed by the band- pass channel available The communication channel used in pass-band digital transmission may be microwave radio link, satellite channel etc. Other aspects of study in digital pass-band transmission are line codes design and orthogonal FDM techniques for broadcasting. Digital Communication Systems 2012 R.Sokullu 337/62

Definitions: The modulation of digital signals is a process involving switching (keying) the amplitude, frequency or phase of a sinusoidal carrier in some way in accordance with the incoming digital data. Three basic schemes exist: amplitude shift keying (ASK) frequency shift keying (FSK) phase shift keying (PSK) REMARKS: In continuous wave modulation phase modulated and frequency modulated signals are difficult to distinguish between, this is not true for PSK and FSK. PSK and FSK both have constant envelope while ASK does not. Digital Communication Systems 2012 R.Sokullu 338/62

Figure 6.1 Illustrative waveforms for the three basic forms of signaling binary information. (a) Amplitude-shift keying. (b) Phase-shift keying. (c) Frequency-shift keying with continuous phase. Digital Communication Systems 2012 R.Sokullu 339/62

Hierarchy of Digital Modulation Techniques Depending on whether the receiver does phase-recovery or not the modulation techniques are divided into: Coherent Non-coherent Phase recovery circuit - ensures synchronization of locally generated carrier wave (both frequency and phase), with the incoming data stream from the Tx. Binary versus M-ary schemes binary – use only two symbol levels; M-ary schemes – pure M-ary scheme exists as M-ary ASK, M-ary PSK and M-ary FSK, using more then one level in the modulation process; Also hybrid M-ary schemes – quadrature-amplitude modulation (QAM); preferred over band-pass transmissions when the requirement is to preserve bandwidth at the expense of increased power Digital Communication Systems 2012 R.Sokullu 340/62

Remarks: Linearity M-ary PSK and M-ary QAM are both linear modulation schemes; M-ary PSK – constant envelope; M-ary QAM – no M-ary PSK – used over linear channels M-ary QAM – used over non-linear channels Coherence ASK and FSK – used with non-coherent systems; no need of maintaining carrier phase synchronization “noncoherent PSK” means no carrier phase information; instead pseudo PSK = differential PSK (DPSK); Digital Communication Systems 2012 R.Sokullu 341/62

Probability of Error Design goal – minimize the average probability of symbol error in the presence of AWGN. Signal-space analysis is a tool for setting decision areas for signal detection over AWGN (i.e. based on maximum likelihood signal detection) (Ch.5!) Based on these decisions probability of symbol error Pe is calculated for simple binary coherent methods as coherent binary PSK and coherent binary FSK, there are exact formulas for Pe for coherent M-ary PSK and coherent M-ary FSK approximate solutions are sought. Digital Communication Systems 2012 R.Sokullu 342/62

Power Spectra power spectra of resulting modulated signals is important for: comparison of virtues and limitations of different schemes study of occupancy of channel bandwidth study of co-channel interference Digital Communication Systems 2012 R.Sokullu 343/62

A modulated signal is described in terms of in-phase and quadrature component as follows: complex envelope Digital Communication Systems 2012 R.Sokullu 344/62

The complex envelope is actually the baseband version of the modulated (bandpass) signal. sI(t) and sQ(t) as components of are low-pass signals. Let SB(f) denote the power spectral density of the complex envelope , known as baseband power spectral density. The power spectral density Ss(f) of the original band-pass signal s(t) is a frequency shifted version of SB(f) except for a scaling factor: Digital Communication Systems 2012 R.Sokullu 345/62

as far as the power spectrum is concerned it is sufficient to evaluate the baseband power spectral density SB(f) and since is a low-pass signal, the calculation of SB(f) should be simpler than the calculation of Ss(f). Digital Communication Systems 2012 R.Sokullu 346/62

Bandwidth efficiency Main goal of communication engineering – spectrally efficient schemes maximize bandwidth efficiency = ratio of the data rate in bits per seconds to the effectively utilized channel bandwidth. achieve bandwidth at minimum practical expenditure of average SNR Digital Communication Systems 2012 R.Sokullu 347/62

The effectiveness of a channel with bandwidth B can be expressed as: bandwidth data rate Digital Communication Systems 2012 R.Sokullu 348/62

Before (Ch.4) we discussed that the bandwidth efficiency is the product of two independent factors: multilevel encoding – use of blocks of bits instead of single bits. spectral shaping – bandwidth requirements on the channel are reduced by the use of suitable pulse- shaping filters Digital Communication Systems 2012 R.Sokullu 349/62

Outline 6.1. Introduction 6.2. Pass-band Transmission 6.3 Coherent Phase Shift Keying Binary Phase shift Keying (BPSK) Quadriphase-Shift Keying (QPSK) Digital Communication Systems 2012 R.Sokullu 350/62

6.2 Pass-band transmission model Functional blocks of the model Transmitter side message source, emitting a symbol every T seconds; a symbol belongs to an alphabet of M symbols, denoted by m1, m2, ….mM; the a priori probabilities P(m1), P(m2),…P(mM) specify the message source output; when symbols are equally likely we can express the probability pi as: Digital Communication Systems 2012 R.Sokullu 351/62

signal transmission encoder , producing a vector si made up of N real elements, one such set for each of the M symbols of the source alphabet; dimension- wise N ≤ M; si is fed to a modulator that constructs a distinct signal si(t) of duration T seconds as the representation of symbol mi generated by the message source; the signal si is an energy signal (what does this mean?); si is real valued Channel: linear channel wide enough to accommodate the transmission of the modulated signal with negligible or no distortion the channel white noise is a sample function of AWGN with zero mean and N0/2 power spectral density Digital Communication Systems 2012 R.Sokullu 352/62

Receiver side (blocks described in detail p ) detector signal transmission decoder; reverses the operations performed in the transmitter; Figure 6.2 Functional model of pass-band data transmission system. Digital Communication Systems 2012 R.Sokullu 353/62

6.3 Coherent Phase Shift Keying - Binary Phase Shift Keying (BPSK) In a coherent binary PSK the pair of signals used to represent binary 0 and 1 are defined as: duration of one bit fc=nc/Tb transmitted energy per bit Digital Communication Systems 2012 R.Sokullu 355/62

The equations (6.8) and (6.9) represent antipodal signals – sinusoidal signals that differ only in a relative phase shift of 180 degrees. In BPSK there is only one basis function of unit energy expressed as: So the transmitted signals can be expressed as: Digital Communication Systems 2012 R.Sokullu 356/62

A coherent BPSK system can be characterized by having a signal space that is one dimensional (N= 1), with signal constellation consisting of two message points (M = 2) The coordinates of the message points are: Digital Communication Systems 2012 R.Sokullu 357/62

message point corresponding to s1 message point corresponding to s2 nc is an integer such that Tsymbol = nc/Tbit Figure 6.3 Signal-space diagram for coherent binary PSK system. The waveforms depicting the transmitted signals s1(t) and s2(t), displayed in the inserts, assume nc  2. Note that the frequency fc is chosen to ensure that each transmitted bit contains an integer number of cycles.. Digital Communication Systems 2012 R.Sokullu 358/62

Error Probability of Binary PSK
Decision rule: based on the maximum likelihood decision algorithm/rule which in this case means that we have to choose the message point closest to the received signal point observation vector x lies in region Zi if the Euclidean distance ||x-sk|| is minimum for k = i For BPSK: N= 1, space is divided into two areas (fig.6.3) the set of points closest to message point 1 at +E1/2 the set of points closest to message point 2 at – E1/2 Digital Communication Systems 2012 R.Sokullu 359/62

The decision rule is simply to decide that signal s1(t) (i.e. binary 1) was transmitted if the received signal point falls in region Z1, and decide that signal s2(t) (i.e. binary symbol 0) was transmitted if the received signal falls in region Z2. Two kinds of errors are possible due to noise: sent s1(t), received signal point falls in Z2 sent s2(t), received signal point falls in Z1 This can be expressed as: Zi: 0 < x1 < æ and the observed element is expressed as a function of the received signal x(t) as: Digital Communication Systems 2012 R.Sokullu 360/62

In Ch.5 it was deduced that memory-less AWGN channels, the observation elements Xi are Gaussian RV with mean sij and variance N0/2. The conditional probability density function that xj (signal sj was received providing mi was sent) is given by: Digital Communication Systems 2012 R.Sokullu 361/62

When we substitute for the case of BPSK Then the conditional probability of the receiver in favor of 1 provided 0 was transmitted is: Digital Communication Systems 2012 R.Sokullu 362/62

if we substitute and change the integration variable: Digital Communication Systems 2012 R.Sokullu 363/62

Considering an error of the second kind: signal space is symmetric about the origin p01 is the same as p10 Average probability of symbol error or the bit error rate for coherent BPSK is: So increasing the signal energy per bit makes the points and move farther apart which correspond to reducing the error probability. Digital Communication Systems 2012 R.Sokullu 364/62

Generation and Detection of Coherent BPSK Signals Transmitter side: Need to represent the binary sequence 0 and 1 in polar form with constant amplitudes, respectively – and (polar non-return-to-zero – NRZ - encoding). Carrier wave is with frequency fc=(nc/Tb) Required BPSK modulated signal is at the output of the product modulator. Receiver side noisy PSK is fed to a correlator with locally generated reference signal correlator output is compared to a threshold of 0 volts in the decision device Digital Communication Systems 2012 R.Sokullu 365/62

Figure 6.4 Block diagrams for (a) binary PSK transmitter and (b) coherent binary PSK receiver. Digital Communication Systems 2012 R.Sokullu 366/62

Power Spectra of BPSK From the modulator – the complex envelope of the BPSK has only in-phase component Depending on whether we have a symbol 1 or 0 during the signaling interval 0 ≤ t ≤ Tb the in-phase component is +g(t) or – g(t). symbol shaping function Digital Communication Systems 2012 R.Sokullu 367/62

We assume that the input binary wave is random, with symbols 1 or 0 equally likely and that symbols transmitted during the different time slots are statistically independent. So, (Ch.1) the power spectra of such a random binary wave is given by the energy spectral density of the symbol shaping function divided by the symbol duration.(See Ex.1.3 and 1.6) g(t) is an energy signal – FT Finally, the energy spectral density is equal to the squared magnitude of the signals FT. Digital Communication Systems 2012 R.Sokullu 368/62

6.3 Coherent Phase Shift Keying - QPSK Reliable performance Very low probability of error Efficient utilization of channel bandwidth Sending more then one bit in a symbol Quadriphase-shift keying (QPSK) - example of quadrature- carrier multiplexing Information is carried in the phase Phase can take one of four equally spaced values – π/4, 3π/4, 5π/4, 7π/4 We assume gray encoding (10, 00, 01, 11) Transmitted signal is defined as: Digital Communication Systems 2012 R.Sokullu 370/62

Signal-Space Diagram of QPSK From 6.23 we can redefine the transmitted signal using a trigonometric identity: From this representation we can use Gram-Schmidt Orthogonal Procedure to create the signal-space diagram for this signal. It allows us to find the orthogonal basis functions used for the signal-space representation. Digital Communication Systems 2012 R.Sokullu 371/62

In our case there exist two orthogonal basis functions in the expansion of si(t). These are φ1(t) and φ2(t), defined by a pair of quadrature carriers: Based on these representations we can make the following two important observations: Digital Communication Systems 2012 R.Sokullu 372/62

There are 4 message points and the associated vectors are defined by: Values are summarized in Table 6.1 Conclusion: QPSK has a two-dimensional signal constellation (N = 2) and four message points (M = 4). As binary PSK, QPSK has minimum average energy Digital Communication Systems 2012 R.Sokullu 373/62

Figure 6.6 Signal-space diagram of coherent QPSK system. Digital Communication Systems 2012 R.Sokullu 374/62

Example 6.1 Generate a QPSK signal for the given binary input. Input binary sequence is: Divided into odd- even- input bits sequences Two waveforms are created: si1φ1(t) and si2 φ2(t) – individually viewed as binary PSK signals. By adding them we get the QPSK signal Digital Communication Systems 2012 R.Sokullu 375/62

Example 6.1 – cont’d To define the decision rule for the detection of the transmitted data sequence the signal space is partitioned into four regions in accordance with: observation vector x lies in region Zi if the Euclidean distance ||x-sk|| is minimum for k = i Result: Four regions – quadrants – are defined, whose vertices coincide with the origin. Marked in fig. 6.6 (previous pages) Digital Communication Systems 2012 R.Sokullu 376/62

Figure 6.7 (a) Input binary sequence. (b) Odd-numbered bits of input sequence and associated binary PSK wave. (c) Even-numbered bits of input sequence and associated binary PSK wave. (d) QPSK waveform defined as s(t)  si11(t)  si22(t). Digital Communication Systems 2012 R.Sokullu 377/62

Error probability of QPSK In a coherent system the received signal is defined as: w(t) is the sample function of a white Gaussian noise process of zero mean and N0/2. Digital Communication Systems 2012 R.Sokullu 378/62

The observation vector has two elements, x1 and x2, defined by: Digital Communication Systems 2012 R.Sokullu 379/62

The observation vector has two elements, x1 and x2, defined by: i=1 and 3 so cos(π/4) = 1/2 Digital Communication Systems 2012 R.Sokullu 380/62

381/62

i=2 and 4 so sin(3π/4) = 1/2 Digital Communication Systems 2012 R.Sokullu 382/62

The observable elements x1 and x2 are sample values of independent Gaussian RV with mean equal to +/-√E/2 and -/+√E/2 and variance equal to N0/2. The decision rule is to find whether the received signal si is in the expected zone Zi or not. Digital Communication Systems 2012 R.Sokullu 383/62

Calculation of the error probability: QPSK is actually equivalent to two BPSK systems working in parallel and using carriers that are quadrature in phase. According to 6.29 and 6.30 these two BPSK are characterized as follows: The signal energy per bit is √E/2 The noise spectral density is N0/2. Calculate the average probability of bit error for each channel as: Digital Communication Systems 2012 R.Sokullu 384/62

In one of the previous classes we derived the formula for the bit error rate for coherent binary PSK as: Using 6.20 we can find the average probability for bit error in each channel of the coherent QPSK as: Digital Communication Systems 2012 R.Sokullu 385/62

The bit errors for the in-phase and quadrature channels of the coherent QPSK are statistically independent The in-phase channel makes a decision on one of the two dibits constituting a symbol; the quadrature channel – for the other one. Then the average probability of a correct decision is product of two statistically independent events p1 and p2. Digital Communication Systems 2012 R.Sokullu 386/62

The average probability for a correct decision resulting from the combined action of the two channels can be expressed as (p1 * p2): Digital Communication Systems 2012 R.Sokullu 387/62

Thus the average probability for a symbol error for coherent QPSK can be written as: The term erfc2(√E/2N0)<< 1 so it can be ignored, then: Digital Communication Systems 2012 R.Sokullu 388/62

Since there are two bits per symbol in the QPSK system, the energy per symbol is related to the energy per bit in the following way: So, using the ratio Eb/N0 we can express the symbol error (6.37): Digital Communication Systems 2012 R.Sokullu 389/62

Finally we can express the bit error rate (BER) for QPSK as: Conclusions: A coherent QPSK system achieves the same average probability of bit error as a coherent PSK system for the same bit error rate and the same Eb/N0 but uses half of the channel bandwidth. or At the same channel bandwidth the QPSK systems transmits information at twice the bit rate and the same average probability of error. Better usage of channel bandwidth! Digital Communication Systems 2012 R.Sokullu 390/62

Generation and Detection of Coherent QPSK Signals Algorithm (transmitter) input binary data sequence transformed into polar form (non- return-to-zero encoder) – symbols 1 and 0 are represented by +√E/2 and -√E/2 divided into two streams by a demultiplexer (odd and even numbered bits) – a1(t) and a2(t) in any signaling interval the amplitudes of a1(t) and a2(t) equal si1 and si2 depending on the particular bit that is sent a1(t) and a2(t) modulate a pair of quadrature carriers (orthogonal basis functions φ1(t) = √2/Tcos(2πfct) and φ2(t)= √2/Tsin(2πfct) ) results in a pair of binary PSK which can be detected independently due to the orthogonallity of the basis functions. Digital Communication Systems 2012 R.Sokullu 391/62

Algorithm (receiver) pair of correlators with common input locally generated pair of coherent reference signals φ1(t) and φ2(t). correlator outputs – x1 and x2 produced in response to the input signal x(t) threshold comparison for decision in-phase – x1>0 decision for 1; x1<0 decision of 0 quadrature – x2>0 decision for 1; x2<0 decision of 0 combined in a multiplexer Digital Communication Systems 2012 R.Sokullu 392/62

Figure 6.8 Block diagrams of (a) QPSK transmitter and (b) coherent QPSK receiver. Digital Communication Systems 2012 R.Sokullu 393/62

Power Spectra of QPSK Signals Assumptions; binary wave is random; 1 and 0 symbols are equally likely; symbols transmitted in adjacent intervals are statistically independent Then: depending on the dibit sent during the signaling interval Tb ≤ t ≤ Tb the in-phase component equals +g(t) or – g(t) similar situation exists for the quadrature component Note: the g(t) denotes the symbol shaping function Digital Communication Systems 2012 R.Sokullu 394/62

it follows that the in-phase and quadrature components have a common power spectral density E sinc2(Tf). Digital Communication Systems 2012 R.Sokullu 395/62

The in-phase and quadrature components are statistically independent. the baseband power spectral density of QPSK equals the sum of the individual power spectral densities of the in-phase and quadrature components Digital Communication Systems 2012 R.Sokullu 396/62

CHAPTER 6 PASS-BAND DATA TRANSMISSION
Digital Communication Systems 2012 R.Sokullu 397/31 397

Outline 6.3 Coherent Phase Shift Keying - QPSK Offset QPSK π/4 – shifted QPSK M-ary PSK 6.4 Hybrid Amplitude/Phase Modulation Schemes M-ary Qudarature Amplitude Modulation (QAM) Digital Communication Systems 2012 R.Sokullu 398/31 398

Offset QPSK In the example from the previous lecture we had the following time diagram for QPSK: Digital Communication Systems 2012 R.Sokullu 399/31 399

QPSK Equations: Digital Communication Systems 2012 R.Sokullu 400/31 400

Figure 6.6 Signal-space diagram of coherent QPSK system. Digital Communication Systems 2011 R.Sokullu 401/31 401

…translated to a space-signal diagram it looks like this: Figure 6.10 which shows all the possible paths for switching between the message points in (a) QPSK and (b) offset QPSK. Digital Communication Systems 2012 R.Sokullu 402/31 402

So, we can make the following conclusions: The carrier phase changes by ±180o whenever both the in-phase and the quadrature components of the QPSK signal change sign (01 to 10) The carrier phase changes by ±90o degrees whenever the in-phase or quadrature component changes sign (10 to 00 – in-phase changes, quadrature doesn’t changes) The carrier phase is unchanged when neither the in-phase nor the quadrature component change sign. (10 and then 10 again). Conclusion: Situation 1 is of concern when the QPSK signal is filtered during transmission because the 180 or also 90 degrees shifts in carrier phase might result in changes in amplitude (envelope of QPSK), which will cause symbol errors (for details see chapter 3 and 4 on envelope detection) Digital Communication Systems 2012 R.Sokullu 403/31 403

To overcome this problem a simple solution is proposed – delaying the quadrature component with half a symbol interval (i.e. offset) with respect to the bit stream responsible for the in-phase component. So the two basis functions are defined as follows: Digital Communication Systems 2012 R.Sokullu 404/31 404

…translated to a space-signal diagram it looks like this: Figure 6.10 which shows all the possible paths for switching between the message points in (a) QPSK and (b) offset QPSK. Digital Communication Systems 2012 R.Sokullu 405/31 405

With this correction the possible phase transitions are limited to ±90o (see Fig.10b) Changes in phase occur with half the intensity in offset QPSK but twice as often compared to QPSK So, the amplitude fluctuations due to filtering in offset QPSK are smaller than in the case with QPSK As for probability of error – it doesn’t change (based on the statistical independence of the in-phase and quadrature components) Digital Communication Systems 2012 R.Sokullu 406/31 406

Outline 6.3 Coherent Phase Shift Keying - QPSK Offset QPSK π/4 – shifted QPSK M-ary PSK 6.4 Hybrid Amplitude/Phase Modulation Schemes M-ary Qudarature Amplitude Modulation (QAM) Digital Communication Systems 2012 R.Sokullu 407/31 407

π/4-Shifted QPSK Another variation of the QPSK modulation technique In ordinary QPSK the signal may reside in any of the following constellations: Figure 6.11 Digital Communication Systems 2012 R.Sokullu 408/31 408

π/4-Shifted QPSK – cont’d In the so called π/4-shifted QPSK the carrier phase for the transmission of successive symbols is picked up alternatively from one of the two QPSK constellations – so eight possible states. Possible transitions are give by dashed lines on the following figure. Relationships between phase transitions and dibits in π/4-shifted QPSK are given in Table 6.2 Digital Communication Systems 2012 R.Sokullu 409/31 409

Figure 6.12 Digital Communication Systems 2012 R.Sokullu 410/31 410

Advantages of π/4-shfted QPSK: The phase transitions from one symbol to another are limited to ±π/4 and ±3π/4 radians (compared to ±π/2 and ±π in QPSK) – significantly reduce amplitude fluctuations due to filtering. π/4-shfted QPSK can be noncoherently detected which simplifies the receiver (offset QPSK cannot) in π/4-shfted QPSK signals can be differentially encoded which creates differential π/4-shfted QPSK (DQPSK) Digital Communication Systems 2012 R.Sokullu 411/31 411

Generation of π/4-shfted DQPSK signals Based on the symbol pair: In-phase component differentially encoded phase change for symbol k absolute phase angle of symbol k-1 Quadrature component absolute phase angle of symbol k Digital Communication Systems 2012 R.Sokullu 412/31 412

Example 6.2 We have a binary input and a π/4- shifted DQPSK. Initial phase shift is π/4. Define the symbols Transmitted according to the convention in Table 6.2 (Formula 6.43 and 6.44) Digital Communication Systems 2012 R.Sokullu 413/31 413

Example 6.2 Digital Communication Systems 2012 R.Sokullu 414/31 414

Detection of π/4-shfted DQPSK Assume that we have a noise channel (AWGN) and the channel output is x(t). The receiver first computes the projections of x(t) onto the basis functions φ1(t) and φ2(t). Resulting outputs are denoted by I and Q respectively and applied to a differential detector, which consists of the following components: arctangent computing block (extracting phase angle) phase difference computing block (determining change in phase) Modulo-2π correction logic (wrapping errors) Digital Communication Systems 2012 R.Sokullu 415/31 415

Wrapping errors In this example θk-1 = 350o θk = 60o (measured counter clockwise) Actual Phase change = 70o but if calculated directly: 60o – 350o = 290o Correction is required. Digital Communication Systems 2012 R.Sokullu 416/31 416

Correction rule: so, after applying the correction rule for the previous example we get: Δθk = -290o + 360o = 70o Digital Communication Systems 2012 R.Sokullu 417/31 417

Block diagram of the π/4-shfted DQPSK detector Figure 6.13 Relatively simple to implement Satisfactory performance in fading Rayleigh channel, static multipath environment Not very good performance for time varying multipath environment Digital Communication Systems 2012 R.Sokullu 418/31 418

M-ary PSK More general case than QPSK Phase carrier takes one of M possible values, θi= 2(i-1)π/M, where i = 1,2,…M During each signaling interval T one of M possible signals is sent: signal energy per symbol carrier frequency Digital Communication Systems 2012 R.Sokullu 419/31 419

s(t) may be expanded using the same basis functions defined for binary PSK – φ1(t) and φ2(t). The signal constellation is two dimensional. The M message points are equally spaced on a circle of radius and centered at the origin. The Euclidian distance between each two points for M = 8 can be calculated as: Digital Communication Systems 2012 R.Sokullu 420/31 420

Figure 6.15 (a) Signal-space diagram for octaphase-shift keying (i.e., M  8). The decision boundaries are shown as dashed lines. (b) Signal-space diagram illustrating the application of the union bound for octaphase-shift keying. Digital Communication Systems 2012 R.Sokullu 421/31 421

Symbol Error Note: The signal constellation diagram is circularly symmetric. Chapter 5: The conditional probability of error Pe(mi) is the same for all I, and is given by: Digital Communication Systems 2012 R.Sokullu 422/31 422

Using the above mentioned property and equation we calculate the average probability of symbol error for coherent M-ary PSK as: (M ≥ 4) Note that M = 4 is the special case discussed before as QPSK. Digital Communication Systems 2012 R.Sokullu 423/31 423

Power spectra of M-ary PSK Signals Symbol duration for M-ary PSK is defined as: Proceeding in a similar manner as with QPSK and using the results from the introductory part of chapter 6 we can see that the baseband power spectral density of M-ary PSK is given by: Digital Communication Systems 2012 R.Sokullu 424/31 424

Figure 6.16 Power spectra of M-ary PSK signals for M  2, 4, 8. OPSK QPSK BPSK BPSK Digital Communication Systems 2012 R.Sokullu 425/31 425

Bandwidth Efficiency of M-ary PSK Signals From the previous slide of the power spectra of the M-ary PSK it is visible that we have a well defined main lobe and spectral nulls. Main lobe provides a simple measure for the bandwidth of the M-ary PSK. (null-to-null bandwidth). For the passband basis functions defined with (6.25) and (6.26) (which are required to pass the M-ary PSK signals) the channel bandwidth is given by: Digital Communication Systems 2012 R.Sokullu 426/31 426

Also, we have from before So we can express the bandwidth in terms of bit rate as: and the bandwidth efficiency as: Digital Communication Systems 2012 R.Sokullu 427/31 427

CHAPTER 3 PULSE MODULATION

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